Is the Radeon R7 250 Good for Gaming?

I use a HD 6570 1GB, which is closest to the r7 240. If you're ok with perhaps 30fps on low settings then go for it. For reference I get MAX 35fps on battlefield 4 lowest settings at 1440x900 (already a 2 year old title). You would obviously get a few more for using a r7 250 over mine.

Twig
 
I have a r7 250 in an old computer I setup for my kids. I just check on the systemrequirments website and it said the video card was good enough for the games you just mentioned. However this doesn't mean it can run them on high settings, it just means there is a setting where they are playable.
As others have said, the r7 250 in an entry level gaming card and if you are serious about gaming it's not going to be suitable.

 
I play GTA V on a r7 260x at 720p, not the best experience ever overclocked it can get 60fps on med settings in the country but when in the city going with a fast car it will drop to around 32fps.. thats with 2GB of vRam 1gb sadly just wont be enough