Thomas
As far as digital multi-meters go, mechanical ones are nicer for reading purposes, but digital ones are nice. They're usually a bit more sensitive than the standard 20,000 ohms per volt mechanical meter. A VTVM (vacuum tube volt meter) is as sensitive as, if not more sensitive than, a digital meter. These are nice and can be found on eBay, swap meets, etc. If you'd like a regular 20,000 ohms per volt mechanical meter, Radio Shack sells a small one that's inexpensive and pretty reliable/accurate. It's somewhere between $15 and $25. For most voltage and resistance checks, this is all that you will need. Only when you check control grid and AVC voltages do you need a more sensitive meter.
Thomas
Regarding the RF section, make sure that all coils have continuity where they should. Make sure that none of the condensers have leakage or shorts. Make sure that all RF tubes are fresh and strong for now. You can try weak ones later once you get the set running. With the band switch, set it for one setting and follow the wiring, contacts, etc. Follow it as it would be connected in the schematic for any one band. This will help you figure out whether it's all connected properly or not. To give you a bit of a hint regarding coils, generally, the higher the frequency range, the fewer the turns on the coil. The wire's gauge (thickness) may or may not increase for the smaller turn count coils.
Thomas