If you don't rent a car, you will probably need to take a bus in Sardinia at some point. They company is called ARST and they have a website which has an English translation option. Its a little clunky but you can do a search for bus routes between cities and for the schedule of a particular line. The buses are modern and comfortable.
After a few experiences, I decided car rental was the way to go. First of all, there are not many busses scheduled and they seem a bit erratic. For example, going from Alghero to Bosa, after 10 am, there were three buses going to Bosa, but only one bus that you could possibly come back.
I was told (and got email from the company) that you can buy a ticket in advance, pay cash on the bus, or use an app. So we wanted to go to Bosa and got there 30 minutes early. Well that bus was 25 minute late, and by then there was a huge crowd waiting to board. I got on the bus and asked to buy tickets but was told they can not be purchased on the bus. He mentioned there was a tabacco shop (about 150 yards down), so I ran down there, and found a ticket machine on the side of the wall - unlike in Sicily where you buy directly from a bartender or a ticket kiosk. I was able to run back and get on.
But on arriving in Bosa, I asked where we get a return ticket and the driver said there is probably a store that sells them. I asked three stores/restaurants and they all said buy on the bus. And again there was exactly one bus returning for the day - and there would be three bus loads of people trying to get on. So it was nerve wracking.
I tried one of the apps recommended and it was a mess. I tried another one, DropTicket, and was able to buy tickets online. When the driver showed up with about 50 people waiting, he said the people with tickets could board first, so I was able to get on, and he accepted the online app purchase.
The next day was a bus strike. Ok, no big deal take a taxi. Hotel says taxis are also on strike. Ended up paying 25 eu to the hotel guy's pal to drive us 15 minutes to the station. Keep in mind the hotel cost was 57 eu for the night, so this was pretty pricey but at least I was able to get moving ... and got a car.