I don't want to act like an expert, played only once two years ago, but, some basics for now are to make sure you have players who qualify for multiple positions(but not over quality), make sure you have as many players as possible playing everyday, make sure your starting pitchers (indicated by the arrow thing) who are pitching that day are in. If you do it right you can get extra pitchers and less position players (which will help your pitching adding stats). As the season goes on, look at the stats you are lacking in and make adjustments. It is pretty random right now, need to let the season go for a month, especially as someone who doesn't watch/know baseball.
Basically there are two types of stats: adding and average. Adding stats are Runs, Home Runs, Runs Batted In, Stolen Bases, Wins, Saves and Strikeouts (Ks). The more people you have playing everyday, and the more starting pitchers you are able to cycle in (relievers pitch more often and more sporadically, so you generally need to stay with 2-3 good closers for Saves only, though they obviously add to the other categories) the better these categories will be. The Average, Earned Run Average, and Walks/Hits per Inning Pitched are all averages, so measure your quality. For instance, if you meet the minimum pitch requirements (there is an inning minimum for your ERA and WHIP to count, don't know what it is) and are either winning big or losing big in Wins and Strikeouts but dominating in ERA and WHIP, you might not start any other pitchers that week, in order to keep those low. You would still want to start your relievers, who will be able to add to your Save category and unlikely to hurt you too bad, as they only pitch one inning most night and starters pitch anywhere from 4-8 realistically.
The way to make this quickest is to do each week at once, cycling through the days to make sure your pitchers are in, etc. You can just hit save at each new day and come back and check it for 30 seconds a day after that. Mondays and Thursdays, once the season gets going at its usual pace in two weeks, are the days teams have off. Usually somewhere around half of the teams don't play those days, maybe a little more. Most other days all teams are playing.