How can I gift an Apple TV subscription?

Subscriptions can’t be gifted specifically. You can gift a generic Gift Card, but technically that can be used for anything Apple.

An Apple TV+ subscription can be changed to annual once the subscription is started. For new subscribers, that can be during the trial period, so paying only the annual price (~16% discount) in advance. For current and previous subscribers, the annual price can start after the current billing cycle.

What would the total be for one year given 16% discount.

The annual price is roughly 12 months for the regular cost of 10 months ((12-10)/12 = 2/12 = 0.167 = 16.7%), often rounded then minus 1, to what Apple thinks are preferable price points (e.g. 99).

In the US, the monthly fee is currently USD 9.99 (before tax). The annual fee is USD 99.00 (before tax).

I’m assuming there are fees / taxes

Possibly. That depends on where it will be used, which you haven’t disclosed yet. In Europe, the consumer price would be already include sales tax (VAT). In the US, taxes differ by state, and the national listed price would not include tax yet for that reason. Some states have 0% on streaming services, others have more.

Though, sounds like if I gave a $100 gift card that should mostly cover the year subscription if they change their billing to annual

Again, that depends on the country/​state where it will be used. $99 +tax might be more than $100.

You don’t.

Apple TV is a hardware box.

AppleTV is a free app that aggregates all video streaming services into one place.

AppleTV+ is Apple’s streaming service.

There is no yearly subscription.

You pay per month and it is tied to your Apple ID.