I think the biggest thing about secondhand is how much you know about bike parts and how much you can do yourself maintenance wise. If you know enough, and have tools, then it's a good option, it's also a good option if you want to learn and invest in tools etc. But if you're just looking to save money then you need to buy carefully. Something may seem like a bargain but if you need to sort new tyres, cables, brake pads, bit of wheel truing, maybe the rims had it....suddenly the price has gone beyond what you could have bought a new bike for. This is a lot to do with how much we pay for parts, I reckon a self build, using new parts, costs as much as 50% more than buying a complete bike.
That said, I do it and thoroughly enjoy it, but I don't do it to save money and, when I make the inevitable errors in purchasing, re-selling the errors loses money as well. Personally I'd make a list that incorporates the intended use with the features/parts that will require, this should narrow down your search criteria. I'd consider:-
weight - are you bothered, do you have an upper limit?
frame material
tyre clearances
frame fittings- do you want to be able to mount mudguards, rack, etc
bars, controls and brakes - what type of each do you want
gears- how many and derailleur or hub gear, if hub gear how do you tension the chain
useage - are you trying for a one bike does all or a dedicated commuter, do you need lighting and, if so, what about a dynamo hub?
Lots of other things to consider but if you can find an off the peg model that matches your needs, then look for deals on last years model, that's probably the best bet, especially if it includes extras that you want like rack and guards.