Check that your iPad is running the latest version of iOS that it supports. If it's very old then it may be limited to iOS 9, and that shouldn't stop you using it. But it's always good to be as up-to-date as possible. Go to "Settings" and hunt around for "Software Update".
When I logged in to TP just now, just above the "Log in" button was a small checkbox labelled "Stay logged in". That's the one you want.
The "Remember me" option you see on a lot of websites should remember your username but not your password. Never seems much point to me.
If step 2 doesn't work, then Safari or whatever other browser you are using on the iPad should have an option to remember usernames and passwords. That won't stop you from having to log in, but it makes the process easier.
Ideally, everyone should use a password manager like 1Password or Lastpass (look in the App Store). These allow you to use proper, secure passwords that are different for every site. If TP gets hacked and you've used the same password for Facebook and your bank, you're in trouble.