What is the recommended replacement for MD5? Since MD5 is broken for purposes of security, what hash should I be using now for secure applications?
That depends on what you want to use the hash function for.

For signing documents, sha2 (e. g. sha512) is considered secure.

For storing passwords, you should use one of the algorithms dedicated for this purpose: e. g. bcrypt, sha512crypt or scrypt. In order to slow down an attacker, these algorithms apply the hash functions many times with an input that is based on the number of the current round. 

Scrypt takes this concept one step further and uses a huge amount of memory. Typical hardware for password cracking has access to about a couple of KB of memory, the default configuration of scrypt requires 16 MB.