I'm a big believer in saving money wherever possible, so I buy almost all of my music gear used, but there are a few things that I have reserves about buying used. I've been a regular seller and buyer on Reverb.com (a great music gear marketplace) for six years, and I've learned a few things. I'd like to share some of those things with you.
What can I buy used?
Usually, effects pedals can be relied on to hold their functionality really well. I almost never purchase brand new pedals because buying them used, usually they look and work just like new, but cost a little bit less.
Also, I've had great experiences buying mid-to-high-end guitars from reputable sellers. I would not buy a guitar from a seller that does not have a history of selling at least two or three guitars beforehand. You can tell a lot about a seller by their selling history. Usually, if you see that they have sold expensive and fragile items and received a 5-star rating based on those sales, you can bet that they will sell you a quality product. If you see that they haven't sold much, it's more of a gamble to buy from them. That's not to say that you can't buy from them, but there's less of a guarantee that the gear you buy will be reliable.
What should I be careful buying used?
Tube amplifiers. Again, unless it's being sold by a very reputable seller that has a history of selling tube amps and packing and shipping them well, it's always a risk to buy these used. Tube amps are very finicky creatures on their own and require proper operation and maintenance to make sure they're working properly, so if you're buying one of these used from a seller you don't know much about, it's a gamble on whether or not that person operated their amp properly and took care of it. Also, there's always the issue of properly packing and amplifier for shipping. Few sellers are willing to ship a used tube amp because they are heavy and there is a large potential to screw it up and damage it during shipping. Make sure you trust whoever you're buying a tube amplifier from and they have a good reputation.
What should I not buy used?
Generally, I wouldn't recommend buying low-end and entry-level guitars used. For the most part, buying them brand new isn't much more expensive, and you can almost guarantee that the low-end guitars haven't been taken care of well by their owners. Usually, used entry-level guitars are being sold because someone bought it to learn guitar and didn't stick with it, so you are buying an instrument that was handled by an inexperienced musician that didn't know how to properly maintain the instrument in good playing shape. The other case I tend to find is guitar players that have been playing for many years and have upgraded their gear find an old guitar buried in the closet and decide to sell it since it's so old and (almost certainly) unplayable.
I would also stay away from used cables. There are plenty of cables that are affordable and brand new and there's no need to sacrifice the longevity and performance of your cables for a few bucks, which is the most you'll save anyway.
I also stay away from buying used straps, but that's a personal choice. There's nothing wrong with used straps, I just don't like the idea of wearing a strap that has had some other player's sweat soaked up into it. Plus, brand new straps really aren't that expensive.
Overall, what it really comes down to is this: you can buy anything used if you really trust the seller and they have a good history of selling quality gear. The opinions above are just based on my experience with the used gear market.