You are in luck...I am a bowler! (598 series last night...2 pins UGH!)
Anyway...here is probably what I would do - though this could be more than you are looking for haha.
PK = Primary Key
FK = Foreign Key
tblBowler
---------
BowlerID (PK)
FirstName
LastName
PhoneNumber
(Any other personal info you want to track - you could put handicap here, though I think it would best be calculated in a query along with average so that it's always up to date. USBC number or things like that could go here.)
tblSeason
---------
SeasonID (PK)
LeagueID (FK -If you are tracking more than one league)
SeasonName (Winter 2008, Summer 2008, etc...depends what you want)
tblLeague
---------
LeagueID (PK)
LeagueName
(Anything else about the league you want to track - if you don't do multiple leagues, you wouldn't need this.)
tblTeamInfo
---------
TeamID
TeamName
(Anything else to track about a team)
tblTeamAssignments
-------------------
BowlerID (FK)
TeamID (FK)
SeasonID (FK)
LeagueID (FK)
(I would highlight all of the fields in this table and make them all Primary Keys because you can't have a bowler on a team more than once. But you can have a bowler on more than one team. Basically that prevents any duplicates across the range. This table is only needed if you do multiple leagues so that you can assign people to teams, otherwise you could just do it elsewhere. It will also keep the data on which team a bowler was assigned for each season/league if you do multiple.)
tblGames
--------
GameID (PK)
BowlerID (FK - so we know who bowled this game)
SeasonID (FK - so we know which season they bowled this in)
LeagueID (FK - so we know which league - they could be in more than one...if you used this)
TeamID (FK - so we know which team they bowled this for.)
GameDate
GameNum (1, 2, 3 - for each game in a series)
GameScore
In the tblGames table, you will have 3 entries per bowler per series bowled. Series scores should be calculated with a query, adding up the games.
It is important to have a date for when the game was played as well - this will allow you to calculate averages and handicaps for any date so that you can generate a report which shows those over time so they can see improvement, etc.
So this all depends on how complex you want to get. I know there is premade software out there which does this, but don't know the cost. If you have any questions, just ask. I think I've covered mostly every bit of data you would need. Possibly the only other thing would be a way to track payments for the league.
Probably a few adjustments you'd have to make....need to relate games to teams to get points won, etc. but should be able to do that with the bowler/season/league ID's in the tblGames table.
EDIT
Actually - now that I think about it - you may not even need the tblTeamAssignments - if you put all that info into tblGame, you should be able to use that for any reporting, so I added TeamID to tblGame.