Here's a complete list of the testnets: https://ethereum.stackexchange.com/a/17101/31933
The most popular ones are Ropsten and Rinkeby. Ropsten is more close to the mainnet as it's PoW-based and therefore works in a similar fashion. Rinkeby is PoA-based so it works a bit differently but it's faster.
The block times (and therefore also transaction times) vary greatly in testnets (at least PoW-based) as there is no monetary incentive. For example here you can see some latest Ropsten blocks: https://ropsten.etherscan.io/ . On the other hand, Rinkeby has stable block times: https://rinkeby.etherscan.io/ .
The faucets (at least for Ropsten) tend to be offline sometimes.
If you want to be thorough with your testing, you can test in the following order:
1) Local blockchain (Ganache). Run all your unit tests here.
2) Rinkeby for integration testing (faster than Ropsten)
3) Ropsten for trying to make sure it'd work correctly also in PoW network
I personally skip phase two.