Welcome to the GJ forums. I have a little bit of experience with this, and can add that Wylie has good advice - he helped me quite a bit.
First off, I think changing your service from overhead to underground is a good idea. Why not just have the lateral (underground feed) go directly from the transformer pole to the garage? If you can make the garage the service entrance and you can place the new meter can adjacent to or just opposite the exterior wall from the main panel in the garage, then you don't need to worry about having a separate disconnect. And yeah, the disconnect and meter socket/can are usually separate. Not only does the disconnect protect the energized cable downstream, but it also provides a means to turn it off to allow for work or in case of a problem.
Ditto on Wylie's comment about the panel in the closet. If you can run all the branch circuits back to the garage, that might be better. Make sure all of your splices (if you need to) are in UL listed boxes and that those boxes remain accessible (can't be behind drywall, etc).
Do you know what kind of conduit that 1.5" is that you've got? I'd suggest running THHN/THWN through it, which is copper that's rated for outside and underground, moisture/weather resistant, etc. I just re-did my feeder to the house which was aluminum SER, underground (at least a short stretch) and in rigid metal conduit. I replaced it with THHN in SCH 80 PVC.
Is 60 amps enough for both the barns? Figure you pretty much just have lights, right? Any electric heat? Might want to think about upsizing those panels, if it makes sense, before you get started. That will also determine if 200 amp service is enough. I up'd mine to 320 continuous (400 amp) with parallel 200 amp panels - one for the shop, one for the house. It probably won't make a big difference on the PoCo side, as long as the transformer is big enough, but they should be able to easily check that based upon the pole number/location.
How many circuits do you have in the house now, including those in the closet subpanel?