I've been down this road myself recently. I upgraded my internet service to 1000Mbit/s aka "Google fiber" and found my long Cat5e runs were limiting my speeds to about 350-400Mbit/s
Here is how I set up my network (and how I would set yours up).
For future proofing, I recommend homerunning a Cat7 cable from each building to a single point where your internet service is located.
Cable
http://a.co/btAdIPV
Male Connectors
http://a.co/0FDhn2p
Male Connector Boots
http://a.co/hCRUyig
Female Connectors
http://r.ebay.com/nN1p9E
Crimping tool
http://a.co/4yb088A
I personally bought all of these products for my setup and my long runs are still able to achieve internet service speeds.
I installed 3 of these wireless access points including one outside on my eave to have internet in the yard.
http://a.co/hBIHgcR
You set these up with the same wireless network name using an internet setup page and they perform seamless handoffs between each other automatically. They will also auto detect if your device can run 5GHz (fast, fair range) or 2.4Ghz (slower, best range). You need a "Power over Ethernet (PoE)" switch to provide power to each of these wireless access points since they do not plug in the wall.
This is the PoE switch I used because I home ran everything.
http://a.co/7hKeX43
In your case, I would install one 5 port PoE switch in each buiding to provide some drops in the building and power the wireless access point.
http://a.co/gcvo1ao
Your cat7 line would be the uplink to each switch in each building.
I ran cat6 once in the building.
http://a.co/cqERh0O
Male Connectors
http://a.co/2E53NdE
Male Connector Boots
http://a.co/7nX53op
Female Connectors
http://a.co/b71Q4Wk
Same tool as above.
Hope this helps.