What should I look for when upgrading from a 60Hz monitor?

I'd start by looking at monitors in the 144Hz to 180Hz range. That's a substantial step up from 60Hz, and you don't need to pay extra for a much higher refresh rate unless competitive gaming is a big priority.

Response time matters too, but treat a 1ms rating as a starting point rather than a guarantee. Check reviews for motion clarity and ghosting on the specific model.

Check what your graphics card can handle before settling on a resolution. A higher refresh rate is more useful if your games can run at frame rates that take advantage of it. Look at the performance you get in the games you actually play, not just the monitor's maximum number.

Panel type is worth a look. IPS generally has better viewing angles, while VA tends to have deeper blacks. VA motion performance can vary, so check reviews if fast games are your thing.

OLED can look great, but static images at high brightness over long periods bring a burn-in risk. I wouldn't make it the default choice for this budget.

Don't get distracted by size alone. An ultrawide can be nice for having multiple apps open, but for gaming I'd think about the desk space it needs and whether your graphics card can handle the extra pixels. A regular widescreen may be better value.