Definitely worth doing some research and getting it right. I had repeated back problems and have felt things get much better since my office got me a Corrigo chair. I am about to be off on mat leave and am going to get myself one for home, as I can already feel my back getting worse from being at a laptop at a dining table/chair.
So I got a cheap desktop and adjustable legs from Ikea (so I can make the desk the right height - even though I am tall I found most desks at normal shops are too high compared with what the ergonomics calculators tell me they should be - and what my work desk is). I think many are a good height for writing, rather than typing. If you Google for desk chair ergonomics I'm sure you will find online calculators that will help you figure out whether your desk is a good height. If it is too low work should be able to raise it for you. If it is too high a footrest may be a good solution.
And then I am going to end up spending a lot (GBP 500, which is about USD 700-800 I think) on the chair. Which may be ridiculous, but being laid up with a bad back is bad enough when you're not responsible for a baby, and I just don't want to take any risks. It should last for ages and get a lot of use from me and OH (who gets RSI problems that would also benefit from a really good adjustable chair).
I think the keys elements of the chair for me are having a really adjustable back, that you can either fix (quite upright to support you) or be flexible (so it reclines with you) depending on whether you are typing or on the phone, etc, and really good lumbar support (mine has a pump/valve so you can adjust it, which has been great through my pregnancy as it has really hugged my back throughout even as the curve of my back changed). The Corrigo website also emphasises that they really recommend arm rests, I think because it stops you hunching your shoulders when you are not typing.
Have you seen a physio at all for your back? If so they could be a good source of advice on what to look for in a chair, and what makes to look at?
Well done for looking into this before you end up trapped on the floor for an hour like I was though. It's really not fun, and I had just put my aches before that down to "niggles" and hadn't realised how close I was to a real problem. It's great for you and your employer if you sort this out sooner rather than later. If your employer is a good size they actually may have dealt with this before? Mine called out a consultant to come and measure me up when I had my problems, and then the chair company came out to set up the chair to fit me. Yes, I'm sure not cheap. But cheaper than having you trapped at home with a bad back for a week when things get too much.
Good luck!