Unless you own a power stretcher, I'd leave the carpet laying to the professionals too. It's not that expensive a labor to try to acheive professional results with DIY equipment.

Depending on your kitchen layout and your countertop choice, you can DIY laminate or butcher block. You can do the final install of most of the plumbing fixtures. You can do the final install of most of the lighting. You can do a lot of the insulating. Actually, you can do the whole build if you have the tools and the time, (we did)but if you're working with a GC who has a schedule, you may not be able to do ANY of it. GC's don't like to work with DIYers. They take a lot longer for jobs and generally don't do the professional quality job that they are depending on and can generate a negative domino effect by that.

Interior painting...
Be sure to budget for the time to do this. You don't want to be hurried or put it off until later.
Exterior painting...
Again, think about your time.
Carpet installation...
I've installed carpet as a youth and was able to do a good portion of my new home. But, I also contracted our entire upstairs out, just to get it done. (Again, the time thing.)
Wood Floor installation...
No comment.
Tile Floor installation...
No comment.
Landscaping (laying sod, planting bushes, etc.)
This can take years, depending on how much property. I'd definitely take this on, but I enjoy sitting on my tractor digging up stumps.
Installing shutters...
Negligible amount of savings, I would think.

You might want to consider using IKEA kitchen cabinets and assembling and installing them yourself. They can look fantastic (do a search on the kitchen forum) and are easy to work with and high quality. Also way cheap compared to similar cabs. If you want to get really adventurous, look into making concrete countertops.

I would not attempt carpet, but hardwoods and tile are relatively easy. Hardwoods are painless if you get the pre-finished variety.

Unless you have prior experience, leave the carpet install to the pros. You can also save some money by assisting various trades such as the electrician, plumber, hvac by being their grunt if they are working solo and are open to it and if they are working by the hour. It also depends on your schedule if you can be there to assist.

DIY'ers cost contractors money.

Very seldom can you get your work done in a timely fashion and keep the job on time for the contractors other subs.

You also need to factor in the extra time it takes you to get the work done and how much more you will pay in increased financing charges.

You didn't mention what, if any, experience you have. The more experience you have already learned while working with a knowledgeable pro, the more likely you can do a better-than-average job.

If you thinking about doing any of the listed projects, but have NO experience, I would be extremely hesitant. If that is the case, sign on as helper to a tolerant GC; you'll learn, without all the stress of being ignorant-and-on-your-own.

None of the projects listed are complicated, but all have their individual complexities. You do need to consider how much time each will take -- a novice takes from three to ten times as long as even a slow pro to complete simple projects (you just can't get up to speed without practice). Sometimes it will be worth it to pay a pro -- not just for his knowledge, but for his speed.

In order, and based upon the unfounded assumption that you have some DIY experience, youth, strength, and are quick-learning:

Interior painting ~~ very do-able, with excellent savings potential. Must be have walls ready for paint, or learn how to properly prepare walls and trim. Not as excessively time-consuming as many projects, but with a high likelihood of extra time spent both prepping and cleaning up. Best started under the supervision of an experienced pro because prep work is 3/4 of the job.

Exterior painting ~~ same as for interior painting, with the additional qualification that special equipment (including scaffolding) is likely to be needed for safest work in high or awkward places.

Carpet installation ~~ in general, not recommended for DIY; with the exception of doing very small rooms IF you have the right equipment and AFTER learning how a pro does it. Labor savings is comparatively small. Many carpet sellers build in the cost of installation -- and don't deduct in proportion.

Wood Floor installation ~~ fairly good savings, more savings in complicated layouts. Reasonable easy to learn; extremely hard physical work (not in lifting but in constant stooping and bending and on knees).

Tile Floor installation ~~ similar to wood floor, but much more skill is needed for proper installation. Must have great patience for tedious work in order to have perfectly aligned andd set tiles; must have basic knowledge of cements and bonding agents for any non-pre-adhesive tiling. It's all too easy for an amateur to be satisfied with results that a pro would consider sloppy.

Landscaping (laying sod, planting bushes, etc.) ~~ Great saving potential. I do recommend using a pro landscape designer for both green- and hardscapes, with you supplying the labor. This is one area where it's worth every penny to supply the grunt labor while taking advantage of someone else's years of study for layout and best plants.
Installing shutters ~~ low to medium savings -- this is a low-cost-for-labor item which depends upon pre-preparation of supports and materials. You will need scaffolding for most installations above ground-level, the ability to use a bubble level accurately, the ability to follow manufacturer's instructions, and the ability to set hinges and drive screws correctly. The greatest outlay for labor comes in priming and painting: the more you do, the less you pay someone else to do. I'd suggest doing the prep work, let the carpenters hang'em.

Not listed but suggested:

Sweeper and trash picker-upper. Don't pay $50/hour to a carpenter for something you can do for free, as long as you do it every time it's needed, as soon as it's needed.

Screw-driver and general gopher. (This is funny, but true.) IF you have a congenial crew boss.... work with and behind the electrician, trim carpenter, and any others doing repetitive jobs. They start, you follow: for instance, support brackets are set and 2 screws started by pro, you follow and tighten each screw, adding more screws if needed for the job. You can install outlet plates by yourself. Be aware that some subs have their own gophers and won't want to save you money by using your bod instead the guy they are used to.

Priming. Almost everything needs to be primed. Some items come with already primed by the factory, but there are still many, many items which will need to be done.
High potential for savings -- this requires labor more than brains.

If you are handy and have tackled some diy in every area of the house, you can save a little money. You can also cost yourself by clogging up the schedule or doing things that will have to be re-done. The best scenario is that you take on jobs that don't have set dependencies. (For example, tiling usually comes before cabinetry, so don't sign up for a tiling job when cabinetry install is tightly scheduled. Painting is pretty flexible in when it gets done.)

Painting is high value, but remember that you're caulking, filling nail holes and doing drywall touchups, as well as all the other tedious prep.

If (and it's a big if) you have an electrician that'll let you do it, you might be able to fish wires. I'd be surprised if the electrician agreed to let you do final hookup of fixtures. And if he did, then you probably get no guarantee, so that might not be such a good deal.

Committing to cleanup at the end of a workday when the crew is gone might be worth money to your general. (About $20/hour here.)

Baseboard and door trim isn't hard, but you need air tools and a decent power saw. Understand that trim carpenters have to cheat a lot to make trim look nice. Do crown last, if at all.

I wouldn't insulate, as in my neck of the woods, the cost of insulation supplied and installed is the same as insulation bought. Plus, insulation is evil.

I wouldn't do carpet (technical), tile (finicky), or hardwood (hard work).

If you were doing the exterior painting and you had scaffold, you could do the shutters at the same time.

If you like landscaping, then go for it.

Finally, I'd note that your time isn't "free". It has a financial cost (time you could be working for someone else) and a psychological/physical cost (being exhausted after a day of regular work and whipping yourself to finish some boring job). On the other hand, there's often a great satisfaction in being involved in building something.

We have done/are doing several things ourselves. We installed the radiant tubing and insulation in the slab, it's not that difficult, just takes a long time.

We've been doing all the cleanup.

We're staining the concrete floors ourselves and doing all the painting.

We'll be installing all interior doors, lighting fixtures, and plumbing fixtures (plumbers did the tubs), and trim.

We're also intalling the tile in the bathrooms.

We're laying the cultured stone with the help of someone who's experienced.

I also stained our front door myself.

I really enjoy the satisfaction of DIY as well as the savings on labor. I love saying "I/we did that!" The cost is being worn out. No wages lost since I'm a stay at home/homeschooling mom. DH works on the house in the evenings and w/e. The kids are getting tired of being away from home so much, but I just remind them that with great sacrifice comes great reward. (and I have taken a couple of days off so we could just stay home).