There are several mechanisms through which a bacteria can evolve antibiotic resistance.
One way is by acquiring an already-existing gene from another bacteria or virus by what is called horizontal gene transfer. In this case, there is no requirement for any mutation to occur although the new gene can be integrated into the host genome and so, in a sense, the genome itself is mutated since it now contains a new gene but the gene itself is not.
Another way of acquiring resistance is indeed by genetic mutations. One beautiful example of this is the resistance acquired toward some aminoglycoside antibiotics. This class of antibiotics interferes with the ribosome assembly by binding to a specific site of it. In some cases, a simple point mutation of the aminoglycoside-binding-site is enough to acquire resistance. Some strains of bacteria exhibit aminoglycoside resistance due to a transport defect (mutation of a channel for example) or membrane impermeabilization (mutation of a pump), acquired by other mutations. Others use specific enzymes to digest the antibiotic, different variants (mutants) of such enzymes broaden the spectrum of resistance.
There are many other different cases in which a simple mutation is enough to acquire resistance.
Here some references:
With the antibiotics removed, is the evolved strain generally more resilient in a normal environment or weaker? No, in general, there is no extra advantage in being resistant to an antibiotic if the antibiotic is not present.