Best places to "spot" locally?
The stretch of road in central Sheffield ("High Street", although nobody I know actually calls it High Street) from Fitzalan Square/ Haymarket/ Commercial Street up to Castle Square is probably the busiest section of road for buses - you'll get a combination of First (Olive Grove) / First (Leger Way, Doncaster)/ Stagecoach (Ecclesfield))/ Stagecoach (Holbrook)/ Stagecoach (Chesterfield)/ TM Travel/ Sheffield Community Transport (plus occasional National Express and all of the trams/ tram-trains), so there's always something passing by.
I'd guess that in a couple of hours, the vast majority of buses in Sheffield will pass through that bit of road (there are a few services that don't go along there but these are mainly the long distance Peak District services (218,271,272,273,274,275 etc), services that don't go anywhere near the city centre (61/62, SL1, SL1A) or the First 18/19.
I've never "spotted" buses with a notebook, writing down numbers or anything like that, but one good place to start would be the various "heritage" liveried vehicles (First have Geminis repainted in 1913, 1963, SYT 1980s, Rotherham and Doncaster liveries plus a B7RLE in the 1990s red/yellow "Mainline" livery). The 51 and X78 are generally the best routes to find these vehicles on (but, as ever with First, nothing is guaranteed!).
Happy hunting!