Really, the *next* thing to do is to get really good at the ship you're trained for. Vindicators do a lot of stuff:
1) Memorize the target priorities for the rooms & sites, so that you can start burning and web down the correct thing as soon as you land, even if FC hasn't tagged yet. Also, *calling* those webs, so that you can coordinate with others.Getting the Ostingeles webbed next to each other so that you can burn 'em all down at once is a fun little mini-game
2) VVV! Learn where the anchor spots are. Learn how to powerslide to a stop 5km *beyond* the anchor spots, so that all those people following you at 5km end up sitting RIGHT in the middle of the next spawn. A good VVV is the difference between a DPS squad that fires at max damage and one that fires at half damage.
3) HHH. Same general idea. Be the guy who sends drones 5km to their next target, not the one who makes them run 50km across the field.
4) DDD. You know the trick where you can make drones chew on a Renyn over there, while webbing & pounding on Schmaeels over here? Can you end a TCRC with a pile of dead Niarjas sitting right at your feet? This stuff matters.
Once you've got all that down... usually if somebody is looking for a second ship type, they'll go for logi. It's a longish train, about ten weeks if you're starting from just the battleship skills, but that just means you've got time to practice that DDD trick