(From your mention of player type and perk slots, I'm guessing you're playing RTTS 2024 or earlier, yes? If not, kindly disregard the following reply.)
Is this your first player that you've made? If so, then leveling up and unlocking the perks will definitely take a while. But it will happen. Thing is, you gotta grind — no simulating. The more you play, the more you'll fulfill the missions on your tier level (X number of hits, extra-base hits, putouts, etc.) and advance your player's development. (Go to your player's page and select "See Tasks and Rewards" tab from the "Skill Set" section on your player's page to keep track of your progress.)
What it really comes down to is that you want to earn those green star things (forgive me if I'm blanking on the term for them just now) by accomplishing missions or succeeding in dynamic challenges or just doing drills and exercises on your days off. Once you have enough, you'll go up a tier (bronze to silver to gold to diamond), your stats will get a huge boost, your Slugging Spark Plug will eventually become The Tool Shed, and you'll unlock more perk slots. Even better, you'll get higher-tier perks which offer you greater advantages — and they'll stay in your inventory, so if one day you decide to create a new player, you can give them the higher-tier perks right from the start, and they'll get to the majors a lot faster than your current one. (For reference, my first player was called up to the Show after about a season and a half or so. My fifth was called up after about three months.)
One other thing you'll definitely want to keep an eye out for is package rewards. The equipment you can get in the package rewards can boost your stats and help you play better, which will earn you more rewards, and so on. (Example: bats improve your contact/power stats, gloves and cleats for fielding/running, etc.) And like the perks, your can get higher tier ones as you improve your player, and they'll stay available for future players you create to use from the start.
Hope this helps!