I would spend more on this Steam sale if I was able to upgrade my four year old

ing GTX960 for a reasonable price.
Do you have the 2GB or 4GB version?
These days, a lot of AAA titles want at least 2-3GB of VRAM (i.e. see Res Evil 2 Remake; AC Unity and newer; Fallout 4; GR Wildlands; Mafia 3; etc etc) - so, if you have 4GB on the 960, you might be okay. Mind you, it won't get the best performance at 1080p if you max stuff out, but at least you'll have the needed VRAM buffer. You should get 30fps+ on most titles (at Medium or better).
Also, I've seen 6GB GTX 1060's around $200-240 online - so yeah, keep your eyes peeled, if you're looking for one of those. Of course, 1070's and above still been quite pricey.
EDIT:
Apparently my computer only has intergrated graphics and doesnt even have a proper graphics card.
Yeah, video cards are
very important these days.
Do you have a laptop or desktop?
If it's a desktop - do you have a PCI-E slot and a decent power supply in there (i.e. 300w or more)? If so, then you could get a solid video card. You're not going to have a decent PC gaming experience w/ AAA titles, if you don't have some kind of solid video card these days. Most modern games seem to want at least 2GB 660's and some even recommend 3GB cards. The upcoming RE2 Remake wants a 2GB 760 for a minimum, so I wouldn't be surprised to see more games asking for mid-range 700 series and above.
If you can't change a lower power supply (i.e. propriety crap, limited room, don't want to change, don't want to deal out more $ for another part, and/or whatever) - look in the
GTX 1050 card range. Your best bet might be a
4GB 1050 Ti for a band-aid (to get well over the usual 2GB VRAM requirement and to get something that does 1080p gaming decently); seen those for around $120 (B-stocks) to $160 (new).
Otherwise, of course, I'd always recommend something like the 6GB GTX 1060 (i.e. $200-240) or better, if you want to dish out more $ and you already have a good system. I have the 6GB 1060 mobile in my gaming laptop and my main gaming desktop has a GTX 970 - and those cards run 1080p @ 60fps or better at High or above on most modern games. If you have a better monitor and whatnot - you could always push to 1440p w/ Medium if need be for 30fps+ performance (often what I do w/ the GTX 970), if you aren't always about having 60fps.