Hey there. These days there are plenty and I mean PLENTY of fantastic monitors for around $200 or less. You won't need to spend $300 unless its for a larger screen size.
The optimal screen size for gaming (or anything for that matter) really depends on your personal tolerance level. Some people can game on a 19" screen while I would never even consider it.
My own personal preference is anything 24" and above. 24" is probably the most common screen size to go with but I would actually recommend you go up to 27" if you can. The extra size takes up more of your peripheral vision and adds to the immersion greatly.
In terms of response time, most monitor these days have very good response times. You will rarely find one that's slower than 10 ms, unless its an IPS. A common response time quoted is 5 ms for TN panels, a bit slower for MVA's and IPS a bit slower than that. ASUS monitors are well known for their quick response times, coming in at around 2 ms (GTG) for many of their monitors.
I hope this has helped you to make a better decision. If you want more info or reviews on some monitors, check out our site.