You are getting a lot of advice from people that are suggesting you do what they did, which is fine for most things, but with plumbing, it can be misleading. Most important thing to consider is how much does your soil move when it freezes? Soil movement is why underground water lines fail. When I lived in CA, and did some underground jobs, there had to be a foot of sand on all sides of the pipe because of earthquake movement in the soil. The inspectors where very specific on this and every job that I was on, there would be well over a foot of sand under the pipe for the inspector to measure. Nobody wanted to fail this!!!!
Black poly becomes brittle over time and it breaks often under ground. I'm surprised that anybody uses it anymore. Here in my area of East Texas, it's no longer used by any cities or water utilities. They made that mistake and they are paying for it now.
PEX is a great product, but it has been known to be tasty to moles and other underground animals. It has enough flex in it to withstand breaking if it freezes, but it's generally not recomended for long runs underground.
PVC in 20 foot lengths with the belled end is a very good, proven product that works great for limited runs. This is where you need to know how expansive your soil is. For 350 feet, this is what I would use. NEVER use blue glue. This is OK for drain lines, but NEVER for water lines under pressure. Here, it's against code and the Water Department will make you remove it if they see that it's used. The only approved method for "gluing" PVC together is with purple primer and heavy duty clear cement. This is a two part process where the purple primer softens the PVC. The clear cement melts the PVC so it becomes welded together. Once this happens, the joint is stronger then the rest of the pipe and if you have a failure, it will be somewhere along the pipe, not the joint. The purple primer is messy, and I believe this is to show the Inspector that it was used. If they don't see the purple primer, you will fail the inspection.
CPVC was mentioned. I've only used it in homes that already have it. I've never seen it in 20 foot sticks, or with a belled end. The belled end on PVC is more then twice as long as any fitting that you can buy to connect CPVC. For this reason alone, I would not use CPVC for any underground supply lines.
Gasketed water lines are the very best option available, and it's the only thing the cities and utilities are using. The pipes slide together. There is a mark on the pipe to tell you how far to put the male end into the female end. This allows the pipes to slide in and out of each other as the soil moves during freeze thaw cycles. This is what I have on my 800 foot water supply line to my house. From what I understand, 2 inches is the smallest available, but I would check to see if smaller was available if you want to go this route. When sliding them together, you have to grease the gasket first. Then you use a board at the other end of the pipe to pry them together. You cannot do this by hand, you need leverage. It's pretty simple and goes together fast enough once you get the hang of it.
Sizing the pipe is based on the size of your meter, the length of the run, and if it's flat, uphill or downhill. Too big of a pipe will lead to stagnant water the remains in the pipe and can lead to a bad smell over time. It's also more money without any benefit. Going slightly larger then the meter size will allow for some friction lost over the length of the run. For a flat run of 350 feet, there is not going to be much of a loss due to friction. If you have a 1 inch meter, you will be fine with a one inch supply line. Gong half an inch bigger might give you some peace of mind, but you wont notice it at the faucet. Most residential meters are 5/8's and the supply line in 3/4 of an inch. The size of your meter is up to your utility company, and what you want to pay. Bigger meter will cost you more upfront, but you also pay more for any additional fees, like replacing the water pump in the well. They base these additional payments on meter size.
