You certainly can start basically any sort of pepper indoors. They are a plant that doesn't seem to mind being repotted, like tomatoes. I'm not sure what your zone is, since WA has many zones, but check out the farmer's almanac website. They have a planting guide, it's more general veggies, so it will just say peppers, but the rules still apply, regardless the size of pepper. You just put your zipcode in and they suggest dates to plant out as well as to start plants indoors for earlier harvests. Here in Colorado, we have a very short growing season, so starting plants indoors is pretty much vital to success lol. They say to start things about 3-6 weeks before your last frost date, but it depends heavily upon the type of plant of course. Good luck!

I would just recommend try different plants and see what you prefer. Some are called "frying peppers" and for "pickling" for a reason, depends on what you want. For bells I like "Golden Summer", very large plants that have light green peppers that are good tasting even before they ripen beautifully. "Flavorburst" is somewhat similar, with light green unripe peppers that are good also. "Big Bertha" is a very nice hybrid to red bell. "Corno di Toro" and "Carmen" are excellent bull horn peppers, sweet peppers at their best. "Giant Marconi" is a very nice long pepper, Burpee's "Big Daddy" is a vigorous ripen to yellow Marconi type pepper.

What do you mean by burning? Given that your are in New Mexico you may want to find a way/place to protect your peppers from the mid day sun. Other than that, choose a variety that is sweet when unripe rather than waiting for one that has to ripen first. Often it's 20+ days between full size and ripe. That's a long time in the sun in NM where things can go wrong. Bianca is sweet from day 1. Also, nothing wrong with seeds from Walmart or wherever as far as germination goes, but Walmart and others do not sell the really good varieties, they sell the common varieties the major distributors are willing to sell cheap. Sometimes you find good veggies, but usually you are better off elsewhere and particularly online where there are scads more choices and much better tasting varieties.

The "white" peppers (those that are cream colored when immature) are very high yielders, but hard to ripen. "Alma Paprika" (mentioned earlier in the thread) is one of those, as is "Feyer Ozon". Both are very compact plants that bear heavily. I've also grown several high-yielding European varieties, "Suptol" and "Taltos". The photo below is of a single plant of "Taltos", pulled just before frost & stripped of leaves: With some reservations, I agree with those that recommend "Cubanelle". It has very large peppers (some 7-8" long) and is incredibly high yielding. So high yielding, in fact, that it requires support to keep the peppers from pulling the plant over, and they are packed in so tightly together that it is hard to pick one without damaging others. Rot can be a problem too, and because it starts from the inside, the peppers might appear normal (for a time) until they are cut open. I've been trying to save seed this year, and it has been frustrating to see so many full-grown peppers begin to change color, only to suddenly turn to mush. To get the full benefit of the huge yield, I recommend picking them green. While I have not grown "Carmen" personally, a gardening friend does, and her plants are loaded, with many ripe. "Sweet Chocolate" (not to be confused with "Chocolate Bell") has elongated bell-like peppers. They tolerate cooler weather, the yield per plant can be huge, and they ripen quickly to a brick red/brown. Thinner walls than a bell though, more like a frying pepper. This one too requires support, otherwise the high pod yield will pull the plant over, or break branches. One of my favorites is another European variety, "Elephant Ear". It has conical peppers 6-8" long that resemble the "Giant Marconi" above, but wider at the shoulder & tapering to a point. The walls are as thick as bells, and the average weight this year has been over 5 ounces per pepper. This has not been a great year for peppers here, but about half of the "Elephant Ear" peppers have ripened, and most of the rest are changing color. I am getting about 8-10 peppers per plant. But for fresh eating & great salsa, I like the "cheese" peppers. They are pumpkin shaped, have very thick walls, and ripen earlier than most bells. "Tennessee Cheese" ripens early, and has a pretty good yield of 2 1/2" wide peppers that are deep red when mature. The walls are often over 1/4" thick, and have less juice than most peppers, so they don't turn to mush when chopped... one of my favorites for adding body to canned salsa. "Greygo" is probably the largest cheese pepper, 4" wide X 2" deep, and average 4 ounces or better. This is my favorite ripe pepper, DW & I eat them like apples... delicious. While the OP is about sweet peppers, even some hot peppers can be sweet under some conditions. That is the case for "Pizza Pepper", from Territorial. The plants are loaded with very large jalapeno-like peppers, 3-4" long & up to 2" wide, with walls nearly 1/4" thick. They can be spicy if harvested during warm weather. When it gets cool though, especially close to frost, most of the immature peppers become almost completely sweet, with a very crunchy texture. Harvested under those conditions, this is my favorite pepper for fresh eating, truly gourmet quality. They also have remarkable storage qualities; I keep them in buckets in an unheated garage for a month with few losses, and in the refrigerator for two months. Over the Labor Day weekend, I visited Heritage Farm (SSE's Iowa headquarters) and walked through most of their trials & preservation gardens. The pepper plot was really interesting - it was how I discovered some of the European peppers that I grow in years past. Saw some great prospects for the future, including "Melrose", a frying pepper sold by several companies. The plants were loaded with 5" long X 1.5" wide peppers tapering to a point, very deep red when mature. It was the last days of August, and not only were nearly all of those peppers ripe, they were shriveling & drying on the plant!!! Early, heavy bearing... looks to be worth trying.