Capri

Go on your own if not in to a tour. Take the hydrofoil.

Capri is very easy to DIY. Just go on your own.

Agreed: go on your own. I do recommend buying a map at the tourist office to help you find your way around. We stayed on the island for 2 nights and didn't have any issues doing our own sightseeing but you should bone up on where things you might want to see are located and if they involve tickets.

A good website:
https://www.capritourism.com

Another agreement - go on your own!

An addition to my post above: you really shouldn't need advance tickets to anything on the island; I meant that some attractions have fees. We were there for 2 days and this is most of what we did:

Hiked up to the ruin of Villa Jovis; no tour needed, just follow the directions. Fee. It takes a little effort but is a great way to lose the crowds and enjoy both some archeology and expansive views. Highly recommend!
https://www.capri.com/en/e/villa-jovis-mount-tiberio

Refuel/rehydrate at Bar Jovis on your return trip from Villa Jovis, it's not far from the ruins and has a nice view!
https://www.tripadvisor.com/Restaurant_Review-g187783-d2298236-Reviews-Bar_Jovis-Capri_Island_of_Capri_Province_of_Naples_Campania.html

Charterhouse of St. Giacomo and Augustus Gardens: Small fees.
https://www.capri.com/en/e/the-charterhouse-of-st-giacomo
https://www.capri.com/en/e/augustus-gardens-via-krupp
(Via Krupp was closed to falling rock when we were there so no go.)

Hiked Pizzolungo to Natural Arch:
https://www.capri.com/en/e/pizzolungo-2

Explored all around Capri and Anacapri. Took the chairlift in Anacapri to the top of Mt Solaro. Fee; highly recommend.
https://www.capri.com/en/e/mount-solaro-cetrella

Assuming you'll be there in season, this was a treat! Pullali Wine bar (right on the Piazzetta in Capri town) has a little balcony that only seats two. We managed to grab it one afternoon just as someone else was leaving, and spent a couple memorable hours just enjoying the great view over some adult beverages and snacks. It can be reserved: I sent a honeymooning couple there for dinner some years back with a recommendation to do that in advance, and he wrote me later that they had had a marvelous evening. One caution: it's right next to the clocktower; the first time it struck it startled us both about out of our chairs!

https://www.capri.com/en/c/pulalli-wine-bar

What we skipped: a boat tour around the island (done plenty of those other places), Villa San Michele, and Blue Grotto. Didn't feel like we missed anything.

I agree that a tour is unnecessary. Do note that Capri is quite hilly, quite big, very pedestrianized, and usually quite hot. For example, the walk to Villa Jovis is very far, and there is no transit option, not even a taxi. So check your interest destinations on Google Maps, with a note of the mileage. And think about hills.

Think about and decide in advance whether the time-sucking, weather dependent Blue Grotto is needed to chop at least 1/4 of the day off your schedule. We skipped it, but I have to admit that we'd previously seen a similar grotto in Croatia. When we met people in the bar of our Sorrento hotel, most had also decided to skip the Blue Grotto, and did not regret it.

Here's a good youtube on the walk to Villa Jovis. While some uphill is to be expected, given the location of the ruins on a high point, we took our time and enjoyed what we saw along the way. We are used to expending quite a bit of shoe leather on our sightseeing days so whether this activity is or is not for every one depends on interests, physical condition and, in our case, desire to escape the crowds in the town centers. It can also depend on time so if you're doing Capri as a day trip, you may not want to devote the time this one takes.

This walk covers stretches from the Piazzeta to Via Longano, to Via Sorpamonte, and then Via Tiberio up to the site. The film finishes before the ruins.

https://www.youtube.com/watch?v=t9STwl3BuZk