I am going to go backpacking for my gap year

Start/finish doesn't matter, and when you go doesn't matter as much so far as you realize it'll most likely be monsoon season depending where you are during the SE Asia stint. Where you go and for how long is completely up to your personal preference and budget. I would definitely do either Thailand to Cambodia and then Vietnam or in the opposite order, doesn't matter. Plenty to do in all three. Could check out Laos too if time permits.

Are you backpacking until school starts in August? Whats your budget?

I would actually start in Europe, do SE Asia, and then finish in Europe to have the best weather for both. The best time by far for SE Asia (as a whole) is November-February. So if you are splitting time even between Europe and SE Asia, I would do October in Europe, November-February in SE Asia, and March-May in Europe. With the you miss most of the Rain and Smokey season in SE Asia and you miss the coldest months in europe. I would check out Holiday-Weather.com for weather info for all the places you want to go. For each city they have "Annual averages" which shows what it is like in each month for things like temperature, rainfall, rainy days, sunny days, etc.

For SE Asia I would start and finish in Bangkok. Do like a clockwise loop. Bangkok up to Chiang mai and pai, over and down through Laos, over and down through vietnam (Hanoi down to Saigon), over to Cambodia, then over to southern thailand and the islands and finish back in bangkok. That is kind of the standard loop a lot of people take.

That being said, try as much as you can to leave things open so you can make plans and enjoy yourself with all the amazing people you will meet. It is nice to have a rough idea of the options in your head going in though.

Cheers!