Should I upgrade from Mac OS 10.7.5 to OS 8 or further up: 9 or 10.10.3?

The first thing that comes to my mind is, upgrading past certain versions may mean your MIDI USB peripherals will no longer work, because later versions require signed drivers, and many vendors won't develop new signed drivers for old peripherals. I believe some 32-bit-only system preference panels may break as well (I'm looking at you, Qmaster). And I think some older apps from the Final Cut Studio suite and older version of Adobe's Creative Suite stopped working as well somewhere between 10.7 and 10.9. The latest version of Yosemite has some DNS problems as well (at least on my box). tl;dr depends on your use case and previously-installed applications and peripherals.

If you can wait until http://www.macrumors.com/roundup/os-x-10-11/, it works well with older Macs.

First create some more space on your hard drive. For any version of Mac OS X you need at least 10% free space. Further you need at least 4GB of memory (ram) The more the better. Before you upgrade alway's have a back-up using Time Machine. Better yet a 2nd back-up in the form of a bootable image on a external drive using Super Duper or a similar app. Once you have taken care of the basics, upgrade to the latest version via the App store. Don't believe the fear mongers about OSX 10.7 - 8 - 9 or 10 that it is causing trouble. This only happens if you already start with a messed up system. If your Mac needs maintenance I highly recommend to do a good http://www.mac-expert.biz/mac-tune-up/ before you upgrade.

Thanks infinitewindow. I'm not even remotely a power user. No Adobe Creative Suite, no Final Cut Studio Suite, no gaming. Just basic Office and Quicken; a printer; an external hard drive. In your opinion, then, to gauge from my basic use of the OS, should I just go all the way up to 10.10.3? (And I'm afraid to ask what DNS means...)

As I understand it, it's pretty involved to get anything but the current version of OS X, i.e. Yosemite (maybe you can talk them into it at a genius bar, otherwise you're going to be torrenting it as far as I know). So, the choice might be made for you. But from your description I don't see any obvious reason not to upgrade to Yosemite.

Thanks, advil!

I concur with advil.

Thanks LuckySeven and thanks again infinitewindow.

As I understand it, it's pretty involved to get anything but the current version of OS X, i.e. Yosemite (maybe you can talk them into it at a genius bar) I was able to talk them into installing Mavericks at a genius bar. They will probably charge you for it, but they'll (probably) do it. There's one downside: if you take this route, you still won't have the ability to reinstall the OS yourself (yes, there will be the "recovery partition", but that naming is a complete lie: it isn't anything of the sort, it's an emergency app store which you still won't be able to buy a version of OS X of your choice from should some emergency arise). If you want backups, you'll need to create your own bootable images. Despite the trouble involved here, I'd still give a "no" to this question: Should I just go ahead and upgrade to OS 10.10.3? http://www.tekrevue.com/yosemite-bugs-time-for-another-snow-leopard/ Yosemite has caused stability issues for every single acquaintance I know that's installed it, and I've heard worse stories around the net. I've also heard of people who have not have problems, though, so maybe you'll get lucky and merely find that you're getting less out of your computer because Apple has decided to spend computing resources older machines have less of on essentially pointless "features." There are some noises indicating that 10.11 might be the stability/performance focused update people have been talking about. I'm a bit skeptical that Apple is constitutionally capable of that kind of focus at this point, but if you can wait, and if your machine seems like it will meet system requirements, that might be a good choice. On the off chance that something might go wrong or you decide that you don't like the upgrade, you should backup your entire current system using SuperDuper or a similar app. Yes. Triple yes.