If your broadband connection is 10Mbps then nothing you connect to it will make internet connections run any faster than that. What connecting the Airport Extreme will let you do is connect your devices using 802.11ac protocol which is faster than the router's 802.11g at 54Mbps. This speed benefit only comes into play for transfers of data across your local network.
should I be connecting my other devices to the Airport Extreme
Try it and see if there is any improvement in performance. Bear in mind that the Airport Extreme supports simultaneous dual band - 2.4GHZ and 5GHz.
do I need a new router that is faster?
If you have the Airport Extreme connected to your router, that is going to give you the fastest possible WiFi connections. With Apple devices (my iMac and MacBook Pro) I can connect to my Airport Extreme at speeds of 1300Mbps. My router (also 802.11ac) only connects at 600Mbps so the Airport Extreme has a much better implementation of 802.11ac than many other devices.