Hi and congrats on your new/old camera!
Any independent repair shop should be able to help you.
I doubt you will be able to get the rear LCD screen fixed... at least not affordably. Replacement parts aren't available, so it would likely have to be removed from a donor camera. But that little screen is of pretty limited use anyway. Even on newer cameras with bigger, higher resolution screens I turn mine off and don't rely on the image playback for anything other than composition and, zoomed in, to check focus. The rear LCD screens on any DSLR are heavily influenced by ambient light conditions... so can't be trusted to judge color rendition or exposure. For the latter, learn to use the histogram. It's more informative.
My first Canon DSLR (after using 35mm film cameras for several years) was the 10D, which immediately followed and is very similar to the D60. Same sensor, same AF system if I recall correctly. 10D was the first Canon DSLR to sell for under $2000 (one whole dollar... $1999 was the list price). If I recall correctly the D60 was the first DSLR to sell for under $2500! (List price $2499).
At one point the rear dial on my 10D was very stiff operating. I used a vacuum and was able to dislodge a grain of sand or something that had gotten under it... and it worked fine after that. Might be worth a try. It also might be stiff from lack of use...Or perhaps the locking switch isn't fully disengaging... operate it a few times back and forth. The "On/hockey stick" position allows the rear dial to function... while the "Off" locks it to prevent accidentally turning the dial and changing settings when carrying or using the camera.
Assuming you get the camera to work well enough for your purposes, another thing you might want to get and can find dirt cheap is the BG-ED3 grip (which 10D also uses... maybe D30, too). It's a very well made and high quality vertical/battery grip... IMO one of the best Canon has made since Power Boosters for some of their film cameras. The reason I recommend this is because those early Canon have two "problems". First, they are slow to wake up from sleep mode. Initially I missed a lot of shots with mine and, for that reason, so deliberately set the camera to have a long delay going to sleep. But that's hard on batteries... and these early cameras already weren't terribly efficient, got fewer shots per charge than later ones did with the same BP511 batteries (and is probably why the camera came with a couple backup batteries). If you need them, additional BP511/511A are cheap, too. I once bought a handful of them for $6 each! Generics that worked just as well or better than the more expensive OEM batteries. Limiting image playback as suggested above also helps conserve battery power too.
Be careful removing and inserting Compact Flash cards in the camera. One of the most common damage I've seen over the years is bent pins in the memory card socket. But with reasonable care I never damaged any of my cameras that used CF cards. I also seem to recall that my 10D was limited to relatively small memory cards and used 1GB and 2GB in them.
If you choose to shoot RAW files with the D60, I think you'll find Canon software no longer supports the CRW files produced by the early models. But there is quality third party software such as Photoshop Elements 2018 that I am fairly certain can handle those files with ease.
The 6MP files produced by D60 and 10D are quite nice. The following image is one I made with my 10D that prints well to 8x10 or even 11x14:
I made as large as 12x18" prints from them, but those smaller files don't tolerate much cropping... So it's important to "fill your viewfinder", if necessary with longer telephoto lenses or by "zooming with your feet".
The AF system in D60 is the same, simple, easy to use 7-point as was used in 10D too. It actually dates back to Elan 7/EOS 30 film camera models, if memory serves.
Like the 10D, your D60 and the earlier D30 also cannot take advantage of Canon's EF-S lenses. Those camera models pre-dated the first of those lenses, which were introduced alongside the "Digital Rebel" or 300D - the first DSLR to sell for under $1500 - which came along a little after the 10D (using the same sensor and AF system as 10D & D60). The good news is that there are lots of third party "crop only" lenses that will work just fine on your camera.... Sigma, Tamron and Tokina made them then and still make them today. Wanting a wide angle lens to use on my 10D, I got and was very happy with a Tokina AT-X 12-24mm "DX" (it replaced and reminded me of a Canon 17-35/2.8 L-series that I used with my film cameras.... but just wasn't wide enough on my 10D). I've also now got a Tamron SP 60mm f/2 Macro/Portrait "Di II" lens that would fit and work fine on the D60, etc.
EF-S lenses were one of the secondary reasons I eventually upgraded to 30D, then 50D and 7D.... and now to 7D Mark II cameras. But at the time the main reason was because the 10D (and earlier models) could not shoot RAW + JPEG.... it could only shoot one or the other. The work I was doing at the time really demanded both... JPEGs for immediate use, but also RAW for later, more refined purposes.
The 10D - and I'm sure the very similar D60 - is by no means a "speed demon". It doesn't have a fast frame rate, large image buffer or a particularly high performance AF system. But I still found it more than adequate for sports and wildlife photography...
I'm sure you'll find the same to be true with your D60.
Hope this helps!
EDIT: I just looked at a comparison of 10D vs D60 online and noticed that the earlier model uses a 3-point AF system... my mistake. I thought it was the same 7-point as was used in the 10D. But I generally used only the center point with 10D (and with later models), so don't see that as much of a concern. Other differences... apparently the D60 has quicker start up, so might be less of a concern than I mention above.10D got an added LCD display on the upper right hand shoulder, which helps when setting the camera. 10D also has higher selectable ISO, though I don't recall ever using it above ISO 800 (due to image noise), but since they use the same sensor and likely have the same image quality I wouldn't call the availability of higher ISOs that might not be usable mush of an "upgrade". But otherwise there aren't a whole lot of differences between them.