Gatorskins are terrific tires for commuting; excellent flat protection. A little "thick" or slow for racing.

Part of the answer though depends upon your trainer, is it "wheel on" or direct drive? If "wheel on" note you'll burn up any regular tire sooner (gatorskins or not). Ideally you'd have a second wheel with a trainer specific tire you could use just for indoors, else you might consider a cheaper tire you can replace more frequently. This doesn't matter if your trainer is direct drive.

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tricalculator.com

First, you have to find out how wide your current rims and bike can handle. You might be able to do that online with a simple search, but you can also take the bike to the shop and ask them. A simple tire swap on your current rims to a much larger size isn't always possible. For example, my work bike (cross bike) came with cheap 700x23mm tires which I soon swapped out for Bontrager AW3 700x28mm tires because they were on a clearance sale, and I thought I could get quite a bit of mileage out fo them. That was the max size those rims could handle even though the gaps on my fork and frame could easily handle much wider. It is a pretty fast tire for my work bike, has good puncture protection and has good traction in wet or dry conditions on sidewalks and asphalt. However that tire is slicker than snot on a door knob on any metal surface such as a manhole cover.

I'm 5'9" and 175lbs/ or 177cm/80kg so if you are about the same size good 28mm wide tires from any well-known brand should work for you provided you ride responsibly.

https://roadcyclinguk.com/...#Usb3rRijKP8c2qIz.97

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The more people I encounter the more I love my cats.

Go airless.... never deal with a flat again. biking is my job. if I have a flat, im not getting paid. These are on my work bike. I just put them on my endurance roadie race bike too..

https://tannusamerica.com/collections/road