I have an AirPort Extreme and a linksys router. What is the best way to get 6-8 wired connections with ac speeds?

The answer depends on whether the AirPort Extreme is your main router....or...the Linksys device is your main router.

In either case, the two devices must connect together using a wired Ethernet connection using CAT5e or CAT6 cabling.

Any version of the AirPort Extreme produced within the last 5 years will have Gigabit Ethernet ports. In order to get the same speeds from the Linksys router, it will also need to be Gigabit Ethernet capable.

If you need additional Ethernet ports, you will need to add a Gigabit Ethernet switch to either the AirPort Extreme or Linksys router.

It is assumed that your other devices connecting to the AirPort Extreme and Linksys router will also have Gigabit Ethernet capability.

Not sure why you mention "ac" speeds in your post. This relates to wireless connections, not wired.

So to get the speeds, do you need to have the airport extreme first in the chain or does it matter?

You have to have Gigabit capable routers at every point in the chain if you want to maintain Gigabit speeds to the devices at the end of the chain.