Cleaning it, don’t be too aggressive, and don’t start by stripping it down completely! The reason for not stripping first is incorrect assembly mistakes can be a pig to diagnose.
Get the worst off with plastic kitchen scourers, tooth brush, maybe brass brush and WD40/paraffin. Once cleanish apply plenty of lube and make sure everything that should moves freely. Ask again if anything is seized, or there’s deep rusting.
If the lathe runs, spend some time putting it through it’s paces by cutting metal. This will reveal faults like nothing else – worn bearings, half-nuts, bed, gibs etc, broken/missing parts, knackered motor, bent lead-screw and other nuisances. Also, if new to lathes, take time to get used to the machine – don’t rush it, there’s a lot to learn.
You can dismantle simple stuff like the tailstock but approach anything else with caution. Lathes aren’t exactly rocket science, but they do have gotchas! Much safer to do a deep clean and paint job later after becoming familiar with the lathe.
I don’t care if my lathe is grubby. Matter of taste!
Dave