Is it bad practice to prefix my hash with the algorithm used?

Many different password hashing systems do this. The type of hashing mechanism used should not be (and should not need to be) a secret. Kerckhoffs's principle says:

A cryptosystem should be secure even if everything about the system, except the key, is public knowledge.

So, if your system is properly secure, it should not matter if the hash mechanism is exposed.

I agree with schroeder that this is OK to do. Even without a prefix, an attacker can probably figure out what algorithm you are using. And anyway, it is the strength of the hashing algorithm that protects the passwords, not the secrecy of the algorithm.

Note that many hashing algorithms and libraries already do this for you. They embed all the relevant information - algorithm, cost factors, salt, the actual hash - into one serialized string. See for instance the Modular Crypt Format or the PHP password_hash function. So don't go making up your own scheme. If you are using a descent hashing library you already got one.

Don't use SHA1 to hash passwords, though. That is not OK.

No, it's not bad practice, and arguably, you should keep this information.

As you observe, this allows you to change algorithm for new passwords whenever you like, without invalidating all users' existing passwords (doing that tends to make you unpopular, for some reason).

There's an argument that this allows an attacker to search out the older, weaker passwords to attack first, but you haven't reduced their security at all (assuming Kerckhoff's Principle, that the algorithm itself mustn't need to be a secret).