You absolutely should. There are many issues with MySQL but I'll give you a very compelling one.

Let's say you want to use a database transaction while you're modifying your table structure to ensure if something goes wrong your data is screwed (causing you to have to restore a backup, etc).

With MySQL any table alterations will persist even though data modifications will be rolled back.

With Postgres, if you cancel/rollback that transaction it all goes back to exactly how it was before.

Another great example is the unique constraint enforcement.

In Postgres if you want to swap two unique values between two rows, you just put it in a transaction and do it.

In MySQL, you have to set one of the values to a temporary garbage value, then change the other value, then change the temporary garbage value to what it should've been.

Another example, if you have a column that's INT NOT NULL...

If you insert a NULL, Postgres will give you an error, MySQL will silently insert 0.