KeePassDroid + Dropbox. I use the desktop version of keepass on my computer, and keep the database file in my dropbox folder. Then dropbox does an OTA sync to let me open that same file on my phone (as well as share it to all my other computers). Instant wireless sync access to all my passwords.
I'll offer a second for OI Safe. Very simple and easy to use if all you want is a basic password keeper. Other, more full-featured, options may be available but OI Safe sure fits the bill for me (and it's free).
I really liked Secret! back on my Palm and WinMo devices and wanted to like it on my Epic. But, I couldn't get past trusting a single app to both secure my information and talk to any site on the Internet. (I know, in theory, that makes it easier to back up and share information between phone and desktop.)
I went with KeePassDroid on Epic, KeePassX on Unbuntu, and KeePass on windows. All synchronized with DropBox as mentioned earlier.
I've just got to get used to always being prompted for username/password/URL even when I want to store a record of just account numbers or miscellaneous facts.
I'm an avid LastPass Premium user. I've used it for years, have it on PC, iPad2, S4, I can use it's browser so it fills in as I move through my interests & has a form-filler function as well. I have many passwords & in no way could keep track of all of it. I don't use every feature- it has many options.
It's safe which is most important. Read their explanation on the topic of all options. I found it easy to use. It's always synced & ready to work for me. Good luck!