"Съесть" means to eat all of something. Я хочу съесть это яблоко - I want to eat this apple.
"Поесть", although it's also a perfective verb, doesn't have the meaning of finishing your meal. It rather means "to have something to eat", "to eat a little". So it should be followed by the Genitive form поесть (чего?) - супа, хлеба, каши etc. The Ginitive here is used to express the meaning "some of". If you don't say what exactly you want to eat then the correct way would be "Я хочу чего-нибудь поесть", but in informal speech people often use the Accusative case instead of Genitive - "Я хочу что-нибудь поесть".
Actually your first variant ( Я хочу что-нибудь съесть) doesn't sound bad either. I could say that in an informal situation, especially if I wanted to emphasise I'm really hungry (so I will likely eat all of my meal).
You can say "у меня" or "у вас" to indicate your or the other person's place. You can also add "дОма".
"Можно поесть у меня (дома)". "Мы можем поесть у меня (дома)". - We can eat at my place.