Any way to speed up this connection?

We've had the same internet service through Verizon for 5+ years and while I would love to upgrade to a better service (such as FioS), it's not available right now. Our plan claims we should see download speeds from 1.5Mbps to 3.0Mbps, but when downloading Steam games (or anything else for that matter), I see an average 300Kbps. Video streaming depends on the day... sometimes it loads quickly, other days it takes forever to load.

Our router is an old Westell Versalink Gateway. The router is plugged into a lan network that runs throughout our house. I see similar speeds wherever I'm plugged in, be it directly to the router or any of the lan connections. Wireless is decent, but it gets finicky on the top floor, so I plan to get another router to extend the wireless signal upstairs.

My question is, is there any way to speed up my overall connection? A new router? I've messed with firewall settings and port forwarding with minimal results.

Thank you in advance.