EVE New Citizens Q&A

You buy the skillbooks from the market.

but remember that each race specializes in different weapon and defense systems.

So Gallente use drones, hybrids and armor tanks, whereas Minni tend to use projectiles and a lot of shield tanked ships.

You will need at least the basics of those to comfortably start switching.

That was my first kick-ass ship fit.

Strictly speaking, you just need their respective ship requirements. Since you mention Minmatar ships, you can take the Rifter, for example. Checking it out via "show info" in game (you can find a standard Rifter link by looking it up on the market), then going to the Prerequisites tab, shows you that you need Minmatar Frigate I to fly it. Sooo... that's it.

Non-strictly speaking, there are a lot of other little things depending on the specific ship. For example, the Rifter has bonuses to small projectile turrets (autocannons and artillery). As such, it is most avantageous to fit that type of weapons. To that, you will have to train Small Projectile Turret.

The armor tanking skills you likely have trained for Gallente ships (if you have any) carry over to Minmatar ships, but some Minmatar ships (like the Breacher) are designed with shield tanking in mind, so you may need to train that whole set of skills to fit shield modules.

There are a few other advanced skills (Sensor Compensation) that are race-specific, but in general everything else carries over.

tl;dr, for cross-training you really only need to train the ship-specific skill as mentioned in the Prerequisites, plus train to be able to use the appropriate weapons for said ship.