Keeping every level of minor league pitching in decent shape can seem like an overwhelming task to new HBD players (and, apparently, some veteran players as well). However, it's actually quite simple-- as long as you stay on top of things. Two minutes a couple times per day will save you big hassles down the line.
Here's what you need to do.
Make sure every level is properly staffed: This means roughly 14-16 pitchers per level, with perhaps a couple more than that for single-A levels. Keep 12 or 13 of them active and stash the rest on the inactive list. Then, you can either let the sim manage your pitching staffs for you (Managers Office --> Management Console, then check the boxes for "manage pitching staff" and change the auto-rest setting to 80% or so) or you can do it manually.
I do it manually because it lets me ensure that my actual prospects are getting enough innings and aren't being pitched below 100%. If you stay on top of things (changing it minimally twice a day) and have a properly-staffed team at each level, it takes under 2 minutes total to go through the pitching for every level-- especially if you use the "recommended roles."
So where do these arms come from?
Promotions: These will prevent guys from retiring. Remember that at the beginning of the season, there is no Rookie Ball. Move up all your high stamina guys from rookie ball to Low A, even if they're not going to ever pitch in the majors. Move up anyone who was even decent in Low A (say, an ERA under 6) to High A. Move everyone who was decent in High A to AA. Move everyone who was decent in AA to AAA.
Tryout Camp: These guys, flatly, suck. None of them will ever sniff the major leagues, and quite frankly most don't even belong in AA. They do have a couple things going for them. 1) They're cheap. 2) It's very, very easy to sign a bunch of high-stamina guys very quickly. If you sign 4 for your Low A team and 4 for your High A team, you'll have more than enough arms to cover you. Just sort the list by stamina (the first red rating that looks like an uphill slope) and then pick the guys with the best splits and control from the top of that list.
Free Agents: These are best for staffing your high minors, if you have a need there. Most tryout camp guys will be effectively useless in AA or AAA, so you'll need to sign some legitimate pitchers to fill those roles. At the very beginning of the season, during the free agency re-signing period, make sure you re-sign all of your minor league pitchers who are asking for $55k or less. This is still a relatively cheap price for guys who will likely be able to eat innings effectively. After that, cruise the minor league free agency lists. It's not worth handing a big league contract to someone who's almost certainly going to pitch exclusively in the minors, but $55k isn't too bad for someone with decent stamina.
Note: There are always a lot of guys with staminas of 8 or 6 on the free agency lists. They really won't be of much help in eating innings, so unless their other ratings are good enough for them to reach the majors, you should probably ignore them.
Maintaining healthy minor league pitching staffs is really just all about preparation. Make sure you've got 12-13 pitchers at each level, then stash another 4-6 on inactive. Make sure most of them have decent stamina (minimally 30, preferably 50+). Then just check in on them regularly. It's like maintaining your house: you can either do it a little each day, ignore it until you're ultimately forced to waste a big chunk of time fixing what you should have been doing all along, or just really piss off your neighbors.