In Settings > Safari:
Make sure you have Private Browsing turned OFF, and Accept Cookies 'From visited'.
This will work with most sites that have a remember me option (usually a check mark on the sign in screen). How long it lasts depends on the site. Some last days, some weeks. They function (track your log in status) using cookies, which is why you need private browsing off and to accept cookies.
You can also select Auto Fill and turn on Names and Passwords. This saves your user name and password for a site after you fill them out the first time. It will help with most other sites.
Nothing works with everything. Heck, even Keychain doesn't work all the time.
There is no equivalent of Keychain on the iPad. The closest you can come is to get a third party password app like 1Password. Some (including 1Password) have a built in web browser than can be accessed directly from the app. You can use it as a sort of secure bookmark list. It's not a great web browser, but it will do the job if you have a site that justifies that level of security.
It is also possible to copy and paste the password from these apps, if you have especially long and hard to remember passwords; which according to security experts we should all have (and they are probably right).
Btw, I use 1Password on both the iPad and iMac. Not the cheapest solution, but it syncs across devices (using DropBox), and is far more versatile than Keychain (which I rarely use these days).