Let me start by saying there are many ways to skin a lawn.

I'll let you know what I do and why (remembering that I'm in a very different climate than you are.)
Why-I'm impatient and lazy therefore I don't like to dig till or weed much. Also, here in FL nematodes can be a big problem in the native soil so doing raised beds without any native soil kinda sidesteps the problem. Oh, and I also don't own a tiller.
What I do-Having just moved into this house I was eager to get a garden started. (owners before me used lots of chemical fertilizer, no mulch or amendments, and the ground is sand.)
I plunked down cardboard boxes and filled them with the free compost from our county and planted right in them. This made for a quick raised bed garden for the spring season.

Over the too hot to garden summer we preped some new planting areas for the fall garden. The other half who isn't as lazy as I did dig out the grass from a couple of beds and amend with mushroom compost. Myself, I layed out cardboard which got covered/mounded with mushroom compost and then covered with wood chip mulch. This fall we already have been harvesting squash and the cucumbers and zucchini are not far behind (those are planted in the mounded mushroom compost on top of the cardboard.) There is also amaranth, sorghum, pumpkins, and corn growing in similar beds. Tomatoes and peanuts are growing in the dug beds.

I personally like the no dig lasagna type gardening but have not had much luck growing things like carrots that way. Carrots did great in the cardboard boxes filled with compost though.

Check and see if your county waste management offers compost to residents, if so the cardboard box idea could get you quick greens for the fall while you work on the in ground garden. You can usually get tons of sturdy cardboard boxes from a produce market. For big expances of cardboard, try the dumpsters behind a rental center. I also get mushroom compost really cheap since my neighbor works at the local mushroom farm. I also have any tree services I see working in the area dump their chips in front of my house, they are usually happy to find a place nearbye to dump the chips and I have gotten three loads this way.

Things to keep in mind.
1-I've found that mowing right up next to the raised beds is difficult. In the future I will extend the weed blocking paper/cardboard beyond the planting area and make a mulch covered pathway to help keep grass/weeds further from the planting beds. This will also help with the cardboard boxes, to make an extra layer of cardboard under them and extending out and cover with mulch.
2-The cardboard boxes will break down over time and you can pull the sides away and hoe the contents to get rid of weeds after the plants are done.
3-This all depends on the type of grass and weeds you are dealing with. If you have something like bermuda grass, you should probably dig it up and pull it out. I understand it often comes back even stronger if you just try to smother it under mulch and tilling it it might not be the answer.

As to your question about adding amendments. First, have you had a soil test done? You might want to hold off on spending money on many amendments until you know if you really need them or not. (having too much of a nutrient can be as bad as not having enough and something like lime should only be used if your soil is acidic and needs it.) Compost would be the exception, go ahead and add that if you like before tilling if you really feel you must till.

Good luck and I hope this helps.

Thanks. I did get soil tests and the amendments are necessary. But as an update, since we finally had some rain yesterday and last night, and it became apparent we weren't going to get anymore; I went ahead out this afternoon with the tiller.

It is so dry here and at this moment I have no way to water (my well went dry, my cistern is very low, and my city water connection is not completed yet), I figured this might be my best chance to till, with finally a little moisture in the ground.

I've read about the lasagna idea and am probably going to give it a try in another area of the yard, later this fall for a spring planting bed (I have plenty of cardboard boxes saved from my move here last month). But without the water, lasagna wasn't going to work as a "right now" option.

What kind of grass is growing there now?
If it is one of the more invasive species, tilling will only make it more appealing for that grass to grow back, especially if soil conditions are made more conducive to grass growth, very similar to conditions vegetables prefer. The grass is one good source of organic matter and should only be thrown away if it is one of those invasive species. It will be easier to spread these soil amendments before tilling the grass but also it would be easier to cover that area with newspapers and mulch and let the grass be digested by the soil bacteria. You could plant the salad crops right through the paper.