What are good amendments, fertilizers, and other nutrients to mix in with the soil where you are going to plant a Tomato or Pepper plant?

It depends on the soil, and how you like to grow.

* I like to use shredded wood mulch on top of the ground for peppers.
* I've noticed good results with using potassium sulfate with some (not all) tomatoes. Haven't noticed much with peppers there.
* Homemade compost seems to work great (I'm leery of buying compost from others, due to herbicides, salt--for composted manure/urine--and stuff, but I've had good success with buying worm castings from others).
* Peat moss is nice, especially if you have alkaline soil.
* I like to use wood ash for both tomatoes and peppers, which seem to appreciate it (but it can make your soil more alkaline, due to the high calcium content; so, be careful if you use it). This year, we burned a lot of vegetable scraps with the wood (so, I'm curious to see how that changes things); it's kind of hard to compost in the winter here (so, we might as well burn them for extra nutrients in the ash--especially orange peels, which don't compost easily, in my experience). Fruit ash probably isn't alkaline, I'm guessing.

I'm with shule on the wood ash which is very high in calcium. (Amateur) testing of my soil showed some high alkalinity which i have been correcting but good amounts of potassium and phosphorus (which is available in wood ash but in very small percentages), so careful application recommended - but i suppose that is like anything you add to the soil..

It sure does depend on your soil - especially anything like lime or wood ash that alters the pH. If your soil is alkaline, both of those would be a mistake, they would tie up nutrients and make them unavailable to the plants. Our conditions are acidic, so lime, especially dolomite lime, is generally helpful.
It's true that tomatoes are heavy feeders and take a lot of nutrients out of the soil. Dried and/or pelleted chicken manure is a good general amendment to increase yields.
I use kelp meal as a source of potassium and bone meal for P and N. Those are easy for me to get here, but there are lots of alternatives.

I like the pelletized chicken/poultry manure also. Medina organics makes it here, 3-2-3. Lots of chicken and turkey farms in Texas. 40 pound bag is $24. I put some under and around the tomato sets when transplanting. Not touching the transplant, but to where they have to grow into the scattered fertilizer zone around and under the plant. Compost gets worked in each season. I don’t quite produce compost enough on my own so I buy a bit, cheap, like $15-20 for 1/2 cubic yard, but you need a pick up truck to get it.

Later on, I mostly foliar feed the tomatoes. Kelp, fish emulsion, something with some P and K. Kelp takes care of the micronutrients, fish, mostly the nitrogen. Foliar feeding is easier for me to apply and the timing and dose is easier for me to control than trying to work in the correct amount of fertilizer into the soil well into the season. That’s a lot of bend over, mulch disturbing and very sort of guess work business as I see it. Foliar is more immediate and the mixture can be be tailored to the needs of the moment and expected upcoming needs.

If the soil has been taken care of by doing a soil test, following the recommendations, doing all the right things including proper sanitation, weeding and having a good mulching program, all you need is a hole and replace the soil around the plant. Add some water and stand back.

Treat the whole garden, not just the area around the seedlings.

I start with some fairly weak organic fertilizer, the pelleted kind you can buy. I have burned plants before by putting too much in the planting hole, so weaker works better for me (cow, horse).
Another option in a garden, is to fertilize the whole planting area, and mix well with a tool. This works also in containers, if you mix the whole thing with some fertilizer first but it's a lot of work. This way you have a more evenly distributed weak dose than maybe too much in the planting hole.

Good point about how to use it. I too use the chicken manure sparsely and dig it into a big volume of soil, never in the hole. Some chicken products have more N and more tendency to burn than others. Same process with the lime and the kelp meal, and the fresh compost. That kelp swells up with water and on times when the pieces were too large, the tomatoes complained loudly and "made faces" (with their leaves) for the first couple of days, I assume because a root had made contact with some clammy piece of it. They get over it though, no burns. Bone meal, I use a 6-8 oz cup in the planting hole for each tomato, then dig it into the surrounding soil. I have never seen any complaints about root contact with bone meal, even for small seedlings.

While I'm new to this forum I'm not new to growing tomatoes. I can still learn a thing or two and this year I tried something different when I planted out on Jan 1. I fish often and this year I happened to be cleaning some Crappie they same day I was planting the starts outside. My starts were about 10" tall and going into ground, not a raised bed. I dug down a foot deep and tossed in two crappie remains in only two holes, then put a few inches of soil on them and planted as normal. Those two plants far exceeded all others in growth and yield. About 30% taller and noticeably thicker stems. I will always do that going forward.

I remember the stories told to us in school about the Indians when they planted corn they first threw in a fish then a kernal of corn and a bean. It is my understanding that as the fish decomposes large amounts of methane and nitrogen are released and travel upwards to the roots. Ive also seen this when burying an animal... the grass above it is way more green and much taller.

[mention]NMRuss[/mention] [mention]Bob M[/mention] I will have some first hand knowledge this year of using fish as a form of fertilizer. In late Winter, whether it was vandalism or some unusual water quality, but I lost over a 100 good size goldfish and did a mass burial in a 5x8 garden bed that I usually grow corn in. Hopefully that isn't an excess of fish to bury at one time but I will see in August...

I can only thus far attest to Crappie remnants after removing the fillet's. I'll also add that I fish offshore and have an abundance of Blue Fin tuna. I'm not a fan of the bellies so I did plant out two tomatoes over them the exact same way as the others. Both plants went DRT within 2 weeks. Tuna bellies are a NO GO!

I bet we could have a whole thread about the benefits and hazards of different kinds/parts of fish.
Capelin is the top fish fert for gardeners in my part of the world. They break down really fast and never seem to burn the plants, whether used whole or just the guts.
I have buried all kinds of fish waste in my perennial garden, including salmon, cod and mackerel. The fatty fish spoil fast for the kitchen but they sure take their time in a garden hole in the local clay soil. Mackerel I opened the hole a season later and it was still there. Not kidding!

[mention]Bower[/mention] Although I buried them fairly deep, I may have to be careful in turning over the soil, not wanting to be digging up any fish parts. I had mulched the area, so I may just make rows and plant seeds.

@Bower This is how the term "Holy Mackerel" originated.

I can attest to the benefits of animal scraps buried in the garden [mention]Bob M[/mention] . I bury nearly everything from my kitchen, into the garden all Fall and Winter. Steak bones, chicken bones, unsalted/lightly salted meat, coffee grounds, egg shells, toilet paper rolls.... You get the point. By the time spring rolls around, everything has been turned into worm castings. One shovel scoop into the garden bed, and it's worm city!

The plants love it too.

I also mulch with "straw" from my ornamental grasses, so this helps to keep the critters out.

The mackerel was still there a year later? Then it couldn't have been holy mackerel, that would have been gone after three days...

I have never had a problem using carp. I put a ton of them under rows this past month. Well, maybe only 3/4. Double dig into the clay, cut them in half with the shovel so they won't bloat and stink, sprinkle with lime, backfill. Never had a problem with growing over them. I don't see how it could hurt, except that time it did, getting jabbed on a y bone under my fingernail a year or two afterward when turning the rows again. The gill plates on big ones don't break down and could possibly cause a problem with a seedling, blocking roots or something when they get turned up shallow. Same as a rock. Still, lots of nutrients, and better for the native fish when these invasive species are removed from the water and into my garden or compost pile.

Directly in the hole over the years, I have used 10-10-10, epsom salt, or fine ground eggshells. But mostly it is just a handful of compost used when transplanting.