I have a researched answer for you. I also have a 310D. You need to pay attention to the JD specifications. Use fluid that meets the Deere specifications labeled J20A or newer oils J21A. This tractor fluid is used in the hydraulic tank, reverser housing, torque converter and the rear transaxle. It is specifically designed for the JD wet brakes and all. Walmart sells a product called Super Tech, Universal Tractor Fluid, TDH. It meets J21A specifications. I buy it in NS Canada for about $37.00 per 5 gallon pale. Best price I can find. Your front axle if 4 wheel drive takes SAE 90 or SAE 80/90 if you live in a cold place. There is no brake fluid. Brakes are run from the rear transaxle fluid. Your engine takes 15W40 diesel oil. I buy the Shell Rotella brand also from Walmart. I stick to JD filters because I don't trust others as the cross reference charts are not always correct and quality can be questionable. You should get an operators manuel at least. Some fluids are checked with engine running and oil hot and some are checked with engine off. There should be a fluid check sheet on the underside of the small service door just in front of your windshield. Deere dealers can supply you with a operators manuel on CD for about $20.00. A parts manuel is available online for free. Go to
www.deere.com. Click on Construction, Parts, type in 310D for a model and you can see all your parts for free. Know the last 6 digits of your serial number to find the right parts for your machine.
Backhoes are fun and John Deeres are tough but the cabs tend to rust. Mine has a rusted out cab bottom. I am now in the process of rebuilding the floor and fenders.
Hope I have helped. You can email back with more questions. I'd love to help.