There have been several threads about washer and dryer recommendations on this forum. It is tough to make a general recommendation because a lot depends on your personal wants and needs, the location of the washer, your budget and so on.
Generally, I think it is a great idea to replace an old TL with a new FL. FLs are much more water and energy efficient and have the potential for better cleaning performance as well.
Let me ask you a few questions to get a better understanding of what you are looking for.

1. Is your washer going to be located on a concrete or a wooden floor (first floor, second floor or over crawl space/basement)?

2. Do you have enough space for a full capacity FL? The big FLs are quite a bit deeper than traditional TLs.

3. How much money are you able/willing to spend? There are always good promotions and rebates out there so you most likely won't have to pay the sticker price. You can expect to get a 10% to 20% rebate (or better) if you buy both a washer and dryer.

Regarding brands to stay away from... I don't think there is an easy answer to that. Some of the manufacturers who are fairly new to the U.S. market may have a lack of service availability in certain areas. If you are interested in LG, Samsung and possibly Bosch you should check out if there are authorized service technicians in your area.
There aren't as many manufacturers as many people think. Whirlpool owns a lot of appliance manufacturers now and keeps selling their slightly different models under various brand names (Kenmore, KitchenAid, Maytag, etc.) Their large FLs are made in Germany.

It would be helpful to know what features you are looking for before starting to eliminate or focus on certain brands.

"....dryer that leaves what appear to be burn marks..."

If those are actually burn marks, you may have other issues at work. Your dryer should not be getting that hot. Suspect clogged vent plus inoperative thermal overload protector -- both of which are pretty common. Properly operating dryers don't burn things. If that is what's happening, this is a genuine hazard that should be fixed before giving it to anyone.

Looser: The laundry room is in the basement on a concrete floor. I'd love to move it upstairs to the garage entryway (also a concrete floor - it's an addition) but that's not in the plan right now or, frankly, ever (just in my dreams). We have plenty of room down there to do whatever we want-it's a full basement that's only used for storage, laundry and some working out.

I'm ok with shelling out upwards of $2K. If they last as long as my current washer and dryer, we're talking $15/month for the appliances themselves. That's more than fair in my view.

We have three small children so we try to get at least one load washed and dried daily. On the weekends, the washer and dryer can go nonstop during the day to get all the laundry done.

My main goal is to get some energy efficiency as well as some efficiencies within the process. Right now, we can do a load of laundry but the dryer takes eons and eons to dry - that's with everything cleaned and serviced and working like a charm. It's almost like I need two dryers for one washer.

I'd like to be able to have counter space above the units for folding - there's a work counter next to the current washer and dryer but it's rinky dink and was the PO's workbench so it's not someplace I'd like to fold on.

Asolo: I believe after having it checked out is that clothes are getting wedged between the drum and the "shell" (forgive my complete cluelessness about laundry terms) and that's what's causing the marks. The dryer isn't getting that hot and the vents are routinely checked and cleared. Thanks for thinking of that - I'll be sure to have everything cleaned out before I give it to my sister.

You don't say what brand dryer you have but the age and symptom you describe is classic of a bad/worn roller.
You may want to tell your sister to have the rollers and seals replaced before she uses the dryer.

Things to think about before selecting a machine...

how 'available' is service locally??? If you're talking with a 'big box' store... ask what town the service will come out of for the BRAND you are buying... you may have to calll the customer service number for the manufacturer to figure out where the closest wuthorized service is...

do you want a 'heater' which will insure you 'hot' wash is really hot, or to enable a "profile' wash... starts out cold and then heats up tto the desired temp... supposedly the 'best' way to get your clothes really clean... do you want the sanitizing function (towels, underwear, diapers, socks, gym clothes... that type of thing... but alll must be able to handle HOT water... hotter than typical hot wash

Steam... supposedly this helps with wrinkles... (personally I haven't figured out if those with this option here realy use it or like it yet)...

all will use less wter, spin faster, and leave your clothes drier when they come out of the wash... all will require you to use HE detergent, preferably powder... and preferably to leave the door ajar when not in use to help with mold/smell issues...

then... the flooring question, which your answer solves many problems people have... concrete seems to be the 'perfect' answer for the 'vibration' caused by the high spinning... again this is a very construction dependent problem... I personally have no problems with my fl on the second floor of my house... others here have had major problems with vibration...

then... go look at them... looks are very different... are ou swayed by the colors... since they are in your basement, maybe not??? each machine out there looks a little different from the other... open the doors, spin the drums around, open and close the soap compartment a few times... what do you like the 'feel' of???

persoanlly I think the dryers are a non issue... and you may or may not even want to get a matching dryer... the more 'normal' looking dryers are MUCH cheaper and just as functional....

decide what you want for features... then go touch and feel models that fit those features...

remember whirlpool makes a lot of the 'typical' brands (sears, maytag, kitchen aid... and I believe others)... so there you're buying by different features on different machines...

whatever you decide, go online before they install and get a copy of the instructions and print them and make sure they follow them... get your shipping bolts from the installers and any other 'tools' that you should receive from them... and make sure they level the thing properly... and tighten the legs...

good luck

Once upon a time I created a 'questionnaire' for folks like yourself. Hope this link helps. Welcome to the land of laundryholics.

Here is a link that might be useful: Laundry questionnaire

Spending more than $2000 is probably not necessary, but it depends on the kind of features you want.
I would highly recommend a washer with internal heater especially when you have three small children. They can get pretty bad stains on their clothing and bedding. Being able to do a profile wash helps taking care of those.

Personally, I don't think the steam function is necessary, but some people really like their LG Tromm and swear by the steam wash. Whirlpool now also offers steam washers and dryers.

On a concrete floor you shouldn't have any problems with vibration. Just like plumbly22 mentioned, make sure that the machines get installed and leveled correctly (or do it yourself).

The advice to go shopping and look at the quality and appearance of the various washers is also very good.
At this point, it is tough to predict the durability of most large FLs. They are still relatively young. You can detect some differences in build quality by "playing" with the floor models though. Some of the door handles, hinges and detergent dispenser drawers feel kind of flimsy and that's probably not a good sign. Maybe those things will never break though. A good build quality doesn't necessarily mean that you'll never have any mechanical or electrical failures either.
You might want to consider an extended warranty. They are reasonably priced at stores like Lowes or Home Depot.

The wash cycles of FLs are longer, but you'll save a lot of time (and energy) during drying. A delay start feature can come in really handy and eliminates "waiting" for the washer to get done. You can have the washer finish when you get up in the morning or when you get home and then you just need to transfer the stuff into the dryer. You can expect drying times of 25 to 40 minutes for most loads (of course depending on load size and fabric weight).
The higher capacity of most FLs compared to TLs should also reduce the number of loads you have to do.

FWIW, I have a Bosch Nexxt 500plus and I am very happy with it.

I chose by reading this forum, epinions, and Consumer Reports. I chose a Bosch Nexxt 500 (300.00 discounted open-box deal). I get it tomorrow and can hardly wait. I hope I am not disappointed.

open-box ? or returned, unhappy ?

Check the washer to see if there's any water under the front lip of the door seal. Also check the lint filter.

Also check the washer before installation to see if it has it's shipping bolts.

Returned because of second floor vibration issues. My laundry room is in the basement, so this was not a concern for me. I got the same warranty that an unopened machine gets. The machine is very quietly working on its second load at this moment (I just took delivery a couple hours ago). I really like it so far. It is so quiet that it's hard to believe that it is really spinning at the indicated speeds!

How do I check the lint filter? Is this safe for me to do?

Absolutely go with the front loaders if you do not mind paying more. I have the LG Tromm (w/out steam) and I love these machines.
Pros of the FLs: cleans your clothes better and more gently that the TLs (I make the comparison that a TL washing machine gives your clothes a shower, a FL machine gives them a bath), more energy and water efficient, depending on the spin speed, the washer will get most of the water out of your clothes and that will result in less dryer time.

Cons: more $$$, special HE detergent, have to leave door ajar until it dries (but that should be done w/any washer), w/Tromm, I pull out sliding detergent, bleach, Fabric softener drawer to empty out water and let it sit out to dry (you get used to this), also, if you want space above your washer for folding, then you do not want the pedestals. You will have to bend more w/out the pedestals.

ALSO -- if you are forgoing the pedestals, may I recommend that you opt for a brand that locates the controls towards the back of the unit (similar to your top loaders). This is one of those things that you won't think about until you are crouching to touch the controls located on the front.

Good luck!