For a reasonable second hand bike made by a known manufacture look at spending between £200-£350. £30 will get you a POS.
Decent new bikes start around £500 & good high spec your looking at spending £2000+
As for things to check, hubs to see if there is any play in the bearings, they can be ajusted but only so much.
Check all the welds on the frame for cracks or aceident damage,
same for the handlebar any cracks they will need replacing.
Crank, check for play or any odd noises when rotating.
Wheels, make sure they rotate freely (no sticking hubs etc) Also they are not bent.
Stem, check for free movement, if it is stiff, may indicate worn bearings or aciedent damage
Gearing, check that gear change is smooth up & down, sloppy change indicates a worn cassette or chain.