When I play as GM I always play every game out...one pitch at bats. I keep track of some stats, events, trends and other things that the game doesn't or doesn't show. By doing this I feel more in tune with my club and can get a true feel either game by game or over time with any changes that I make (ex. Mgr tactics, individual player tactics, adjusting lineups, bringing in/sending out guys etc.). I am sure that some will say that they can do this without playing the games out. Good for them if they can do that. To each their own.
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I also check the wire every day as well as a quick look at the free agents. I then look at my minor league injured list to keep track of what is going on down below. These things usually take me about 3 minutes or so unless I need to address something. I generally make changes to the minors each month, at the half way point, in the offseason and as needed. (ex. Many signings due to a rash of injuries, promotions/demotions, culling the lowest levels before or after the June draft, retooling in the offseason etc.).
As far as improving your team goes...WOW...we could be here for awhile if we start to go into detail about all the ways that you can or might improve so I will keep this short.
TRADES - Make trades that make sense. If you are a small market team trade your players before they become FA, at least a year before and get players that are still in the minors and are almost ready so you KNOW what you are getting. If you just trade for a prospect, he may not pan out. On the other hand, if you have the money be smart about it. Don't just throw it away. Spend in FA if it makes sense. If it doesn't, then look towards a trade.
CONTRACTS - Be VERY careful about the length and amount of the contract you give out. There is a saying. "There is no such thing as a bad one year contract". What that means, if you are unaware, is that a contract of one year in length, can be as large as you want to make it, as long as you stay within your financial parameters. As an example let's say fictional RF Joe Smith just became a FA. He is the hottest thing going but had a down year last year. He is only 27 and has been very durable and his track record shows that last year was very likely nothing more than a hickup. The teams to offer him longer years are offering a lot less money than what a caliber of player like he deserves, so he declines them all. One team comes along and their offer intrigues him. One year $42.5M. He and his agent think this will be good for him. A "prove-it" deal for that huge amount and then next year he will command much more than he had in the past. He thinks to himself and smiles "somebody should have signed me this year for good money on a long term contract, because next year it's gonna cost them".
Moving on about what to avoid with contracts, watch how many years you offer and be willing to put a cap on how much a player is worth. You will have to learn what those values are as they pertain to your league. In my universes I always limit contracts to 7 years so as to not let the AI hurt themselves.
Always look at the big picture and take the long view when it comes to being a GM of a baseball team. Doing the opposite can hamstring your club and cripple your team for years.
I will leave it here for now. I hope that some of this is helpful to you. I'm sure some of the other great forum members will have some more ideas to pass along as well.
Have a Great Night!
Palaaemon