I'd say the most two important things, other than starting, and running looking for smoke, etc., would be the transmission, and brakes. Make sure it will pull itself in all of the gears. 4th may be a little sluggish, if you don't have room to run. Make sure the shuttle works properly, shifting from forward, to reverse. Be at a complete stop, before changing direction. It has a torque converter in it, and it would be like shifting your car, or truck from R to F, and could cause damage. if you're not stopped before changing directions.
Of course, there is the normal checking of the fluid levels. Look for milky hyd., oil. Pull the trans. fluid dipstick, and look for proper level, and even smell it, to see if it smells burned. And engine oil.
Check brakes, to see if it stops well. These have pretty decent brakes on them if in proper operating order. Brake assemblies are a little tough to see with the cab on. You'll have to get down low, or crawl under the side, just ahead of the rear wheels. The main thing to look for here, is oil seeping below the brake assy. There are usually drops of oil hanging off the bolts, and/or case.
I went to TractorData, to get the lifting specs for the loader. It's the first in the listing here:
TractorData.com J.I. Case 58C industrial tractor attachments information
A 3pt. hoe would probably be better than nothing. It will be a job to put an OEM hoe on it, do a job, then remove, and put the weight back on. The OEM hoe will have a lot more reach, and you may even find an extenda hoe for it. Depending on what you plan to do with it, it may be easier to rent a mini excavator, and have the loader tractor as support, especially if you're putting in tile etc. Trench with the excavator, add gravel if needed, and backfill with the loader. Helpers with experience help here too.