what would be the best way to get internet to my shop?

Given the distance, Ethernet would be difficult to implement. Otherwise that would be the way to go.

I suggest you look into a WiFi router and a WiFi AP of some sort. Run an antenna wire from the router out the wall and mount a directional antenna on the outside of the house pointed at the external building. Search for "Cantenna" You can get commercial products or find instructions to make your own.

You would then place a Cantenna on the external building aimed at the first one and wire it to the internal AP. Alternatively (and preferably I think) you could set up a point to point microwave connection. It would be a little more expensive but a much better/faster connection.

What you should look at is a proper (semi-commercial) wireless bridge. That assumes you have line of sight between the two sites. A popular vendor of bridges is Ubiquity and they are not too expensive (usually less than $100 a pop - a whole bridge would be less than $200). There are other vendors as well. The key is to look for a "wireless bridge", that has a pair of units (one at each end).

You mount those bridges outdoors, pointed to each other. The bridges can provide very high speed links.

Another solution is a POE powered switch, one you can place in a weather proof box halfway (300ft), and then run outdoor rated Ethernet cable just below the ground. You power the switch from one end with a POE injector. However, that may cost almost as much as a wireless bridge anyway....

600 feet is about twice the rated length for Ethernet, it is possible you could run a cable directly, but you might have to use a repeater between the two locations to get a good signal.

If you do, you have two choices, also run a power cord with it, and power it somewhere in the middle, or get a power over ethernet injector and use that. Either would work, and this service could be done by any competent network company, or by yourself, if you're willing to buy the equipment you need yourself and run the cable.

It is also possible to set up a repeater using Wireless, again, depending on your situation, that will also work, but obviously it depends on how much you need quality networing.