You can use a wallet generating site like https://myetherwallet.com. There is a lot to learn. It generates wallets for you. Safely storing wallets are a whole subject on its own. The safest and most secure way is to use an "airgap" (not connected to the internet) computer to generate you wallet and keys so that a hacker cannot get on your computer and steal them. I have installed the myetherwallet.com site on a raspberry pi and then I generate keys on there and print them (cable directly connect to my laser printer) on plastic paper that is waterproof, then I seal them up with anti tampering holographic stickers (so I know if someone has messed with them) and stick them in my safe in various amounts. 1 eth, 5 eth, 10 eth 25 eth. On the outside of the paper wallet (it is called), is the wallet address and inside is the private key. Then I can just grab one and check the balance every so often.
That is a called cold storage. To learn about the different wallet types, check out this site;
About device switching. You should get a new wallet address (back it up, again, I print my backups on an airgap device and print to duracopy paper, place in safe) and put in on the device when ever you switch. Always diversify your accounts. I would use a mobile wallet for transactions in person, with low amounts on it. My big amounts are on cold storage in my safe. If you lose your device you have the account backup in your safe.
If you have your wallet address and keys in your possession then you should not have to worry about the site or application going bankrupt as the address and keys go with you. If you buy some on some site, transfer it to your wallet and verify that it went through. Then you do not have to worry about the site.
I hope this helps.