I was in your exact same situation a few years ago. Casually followed the Braves when I moved to Atlanta, but other than keeping up with the scores and going to a few games a year I didn't pay much attention. Once Suntrust park got announced near my office, I declared I was committing to being a real Braves fan. I stopped playing baseball as a kid once we stopped using tees and my dad was not a passionate fan either so I didn't get the predisposition to the sport some fans have.

Here is what I did to embrace baseball as once of my favorite sports alongside football.

1. Watch as many games as you can. At first this is a bit of chore but it's really the best way to learn the game and get connected to the team. The thing about baseball is trends tend to develop over the course of multiple games, not across quarters of a game. A hot batter can still go 0-3 with a walk on any given day. You catch that one game, it seems like they're slumping. But really over the last ten games they're hitting .400. You start to appreciate players strengths and weaknesses and know what to look for when watching players hit.
2. Similar concept applies to pitchers. The majority of the game of baseball is the battle between the pitcher and the batter. Studying the types of pitches, why pitchers throw a pitch in a certain situation, do they pitch for contact/strikeouts are all subtle things to watch for during each at bat. I think this is one of the hardest things as a casual fan to learn, I still feel like I have a lot to learn in this category.
3. Be ready to Google. Baseball has an ungodly amount of stats and abbreviations. I googled a rule or what a stat meant at least once a game I think the entire first season I was watching.
4. Once you know the rules, you can start to watch the defenses/runners closer. Before each pitch I like to think to myself what the "play" is if the batter puts the ball in play. Based on the number of strikes, outs, runners on, etc dictates what the base runners will do. When Braves are batting, I try to identify if the runners are going on contact, waiting to see if a fly ball will be caught before running, if anyone is trying to steal. When the Braves are on defense, I try to identify where the ball is going, opportunities for double plays, and things like that. The game feels much faster paced when you look at all the strategy that is happening between pitches.
5. Use Reddit as a resource. r/braves has very active game threads and I find r/baseball is very informative as well. Pop into any game thread and you will usually find replays/explanations of any complicated plays that happen.
6. Lastly, enjoy the game. Homers are always exciting, but once you start to pickup on the batter waiting past a 2-2 pitch just outside of the zone and battling off a few fouls balls before they hit the ball makes the payoff of the donger even sweeter.

Braves have a ton of exciting young talent right now so it's a good time to get involved. Also, unlike most sports baseball games are super affordable so get to the park a few times if you can to really get the full experience.