MD5 for passwords
Using salted md5 for passwords is a bad idea. Not because of MD5's cryptographic weaknesses, but because it's fast. This means that an attacker can try billions of candidate passwords per second on a single GPU.
What you should use are deliberately slow hash constructions, such as scrypt, bcrypt and PBKDF2. Simple salted SHA-2 is not good enough because, like most general purpose hashes, it's fast. Check out How to securely hash passwords? for details on what you should use.
MD5 for file integrity
Using MD5 for file integrity may or may not be a practical problem, depending on your exact usage scenario.
The attacks against MD5 are collision attacks, not pre-image attacks. This means an attacker can produce two files with the same hash, if he has control over both of them. But he can't match the hash of an existing file he didn't influence.
I don't know if the attacks applies to your application, but personally I'd start migrating even if you think it doesn't. It's far too easy to overlook something. Better safe than sorry.
The best solution in this context is SHA-2 (SHA-256) for now. Once SHA-3 gets standardized it will be a good choice too.