I am either really terrible (most likely) or it just takes a really long time to finish the conquest. Assuming I'm actually just awful at it, what are your tips on completing it?
Yeah definitely depends on how much time you have to play... the key is to protect your stronghold when you see a threat coming but for the most part continue to take empty territories and put majority or all of your fans in one territory to attack other teams territories with all your other territories maintaining the required 1 million fans... only reinforce those territories if there is a threat of them getting taken.
For the First Inning Conquest you will have to steal fans as it is quite difficult to pull off without doing that plus thats one of the missions anyway... I recommend stealing from the Padres as I believe that is the only specific teams stronghold that you have to take... so I'd work on them first before moving on to the other teams.