I understand that the speed that a website loads is dependent on many things, however I'm interested to know how I can positively impact load speed by increasing the specifications on my dedicated server:
Does this allow my server to handle more requests? Requests come in and are essentially put into a queue until the system has enough time to handle it. By increasing system resources, such a queue might be faster processed, and such a queue might be configured to handle more requests simultaneously, so... yes-ish (note: this is very generalized)
Does this reduce roundtrips? No, your application design is the only thing that effects this. If your application makes a request to the server, it makes a request (e.g., a "round trip"). If you increase your server resources, you do not in turn decrease the amount of requests your application makes.
Does this decrease server response time? Yes, see first explanation. It can often decrease the response times for the same reasons given there. However, network latency and other factors outside the realm of the server can effect complete response processing times.
Does this allow my server to generate pages on Wordpress faster? Again, see the first explanation. This can help your server generate pages faster by throwing more power at the processes that generate the pages. However, outside factors aside from the server still apply.
For performance, the two high target areas (assuming you don't have tons and tons of traffic, which most sites do not), are reducing database reads and caching. Caching covers various areas... data caching on the server, page output caching, browser caching for content, images, etc. If you're experiencing less than desirable performance, this is usually a good place to start.