Novorossiysk is a beautiful city with a very unusual and majestic embankment, so why not start your Crimean trip from there? The bus costs about 600 rubles per person, travels about 5 hours.
There's nothing to do in Kerch, though - it's sad and Soviet. But the ride over that long bridge is a very memorable experience. Interestingly, we still have to cross the border, even though we are traveling through Russia. Apparently, this is because Crimea is a Republic.
Inside the Crimea you can also travel by bus from 100 to 500 rubles, from 30 minutes to 4 hours between cities. This is very convenient, so even if you do not have a car, you can easily get from one point of the Crimea to another.
If you drive a car, I'm sure there are other laws at work here, which I, as a pedestrian, find difficult to reason about.