Does a Faster internet speed equal less lag?

Generally speaking yes, but bandwidth is not just about speed, it's about how much data can be sent at once, which obviously affects speed, so the more bandwidth you have generally the faster the connection is.  That being said there are other things that can slow down your internet, bad hardware, bad physical connections, how you are routed by your ISP, etc.

PING is the best indicator of pure speed. PING is a speed test where 1 packet of data is sent to another host - the time it takes to get there and respond is measured.... 20ms for example, this is where the term LAG comes from otherwise known as LATENCY.  The higher the ping the more lag you have.  So if your ping is 30ms or less, that's considered good.  Under 10 is really good.  Over 40ms gets dicey, and so on.  This is why playing on a server close to you is so important, less hops to get there.

30 mbps might be just fine for gaming, you need to test your latency.