I grew up in Al Ain for 15-or-so years, so I decided to compile a list of the UNESCO locations in Al Ain, categorised by type, with additional locations in and around them that may be of interest.
Locations in bold are part of the inscribed UNESCO WHS (on the official website) and those in italics are not accessible using Al Ain's bus system (so you'd have to drive or take a taxi). Some may require a 20-minute walk from the nearest bus stop, so be mindful of the heat!
I no longer live in the UAE so information could grow outdated over time.
SUMMARY (TL;DR):
Al Ain Oasis for an oasis
Al Ain Palace Museum for some royalty
Beehive Tombs for tombs
Hili Archaeological Park for more tombs
Qasr Al Muwaiji for a fort with a museum
Jahili Fort for a fort with cultural events (check visitabudhabi's Events page)
Mubazzarah and Jebel Hafeet for a relaxed day out
Qattara and Jimi Oases for more secluded forts, more oases and more events
Check to see if Bidaa Bint Saud ever opens. I'd be curious to know
OASES:
Al Ain Oasis - The main, central oasis. Free to enter (as is all oases), it not only provides examples of Aflaj (ancient irrigation system, like in Oman's WHS), but it acts as a shortcut between Al Ain Museum and Al Ain Palace Museum, all while the palm trees shade you from the scorching Sun, its thin prongs allowing some sun-rays to gleam through, creating a heavenly feel.
Al Ain Museum - A small museum with a small fort.
Al Ain Palace Museum - Sheikh Zayed's home (founder of the UAE). Good sample artefacts, like an open-air museum.
Jahili Fort and Park - Pretty empty. Look out for cultural events here on visitabudhabi's Events page. The park is a decent picnic spot if you cannot make it to Green Mubazzarah.
Qattara Oasis - Aflaj can be found here too. Has a decent number of forts in and around it. Most forts surround oases to protect them, their resources and their farmers, which is why there's so many of them.
Al Darmaki Fort and House - The house's ruins are surprisingly striking. Most forts have been reconstructed so it's a nice change.
Qattara Arts Centre - Local art venue, so look out for events here as well (visitabudhabi). Has a small souq (market) where Emirati women sell crafts. And of course, there's a fort there.
Al Wajidi Fort - Yet another fort. These forts were mostly made with mud, camel dung and straw.
Jimi Oasis - Nowt special. Qattara and Jimi oases have many useful throughways just like Al Ain Oasis.
Bin Helal Fort and Heritage Masjid - I think you get the point. There are many restored or reconstructed forts in Al Ain.
Bin Hamoodah Fort
Al Jimi Fort
Hili Oasis - It's quieter and further out, but Al Ain Oasis isn't really packed with people, so there isn't a whole lot of difference.
Hili Fort - Nothing special but is at the entrance of Hili Oasis.
Bin Hadi House - In the middle of Hili Oasis. Ruins originally built in 1820s. I haven't seen it for myself.
Rumailah Fort (Near Rumailah Site too) - Nonagonal fort but cannot enter.
Hili Towers - One's a cuboid, one's a cylinder!
Muwaiji Oasis - Can you actually enter this oasis? It looks like private farms to me.
Qasr Al Muwaiji - Birthplace of the UAE's second president. Provides the best visiting experience of the forts (the others were basically empty based on my memory), but is reconstructed too perfectly for its own good.
Mutaredh Oasis - No forts around here as far as I know. I haven't been here.
PREHISTORIC SITES:
Jebel Hafeet Beehive Tombs - Similar to those in 'Bat, Al-Khutm and Al-Ayn' tombs in Oman. The Hafeet mountain range provides a looming backdrop. Apparently you can go dune-buggying here too.
Hili Archaeological Park - Whenever my Dad and I visited, the park was women-and-children-only and manned by a security guard at the gate, who would let us in as an exception. Rubbish around the place, including within the enclosures. The Grand Tomb, with its characteristic oryxes is the main landmark. Worth a visit.
Hili 2 - In private property. A policeman that just happened to be passing by saw me tip-toeing over the wall to take a picture and politely told us it was private property. I couldn't really tell what I was looking at in the photo anyway.
Bidaa Bint Saud - To my surprise it actually seems reachable by bus. It's fenced off though, and the ruins are atop a hill, so you cannot see them from behind the fence.
Naqfa Ridge - Went here and saw no ruins. I do think it's quite cool how there is this very slim ridge that cuts into the city and winds like a spine. There's a viewpoint there too.
Saniya - Want to get an idea of the grimy, industrial side of the UAE? Take a walk around here.
Khalifa Bin Zayed Stadium - If ever you wanted to watch Al Ain FC.
Rumailah Site - In private property, and I could barely see anything over the fence. Ruins are hard to make out.
Hili North Tomb A + B - I haven't been here, but judging by satellite map, they appear to be in a private farm at the back of a theme park. Might as well visit Hili Archaeological Park.
West Ridge Tombs - Tombs that look like they're in some industrial site. Perhaps not the best idea to try to get to them.
Al Ain Wildlife Park Tombs - I don't know at this point.
Jebel Hafeet North Tombs - I wonder if they're considering making these unknown locations visitable.
NOT INSCRIBED (Additional Places to Visit):
Cultural Events - Mainly in Qattara and Jahili Fort. Check on visitabudhabi's Events page for anything. Places in Al Ain may be labelled as 'Abu Dhabi'.
Green Mubazzarah (Jebel Hafeet) - Peaceful green park in a jagged, rocky landscape at the foot of Jebel Hafeet. Great place for a picnic.
Mubazzarah Dam - Built in 1955. It is walkable from Green Mubazzarah but it takes half an hour.
Jebel Hafeet Summit - Pretty mediocre views. Just vast flat desert (not even dunes!) with the odd industrial area. Might as well take the drive up while you're at it. There are a few hotels there too.
Al Ain Zoo - Houses a natural history museum alongside all kinds of animals, including the local oryxes
Sheikh Khalifa Bin Zayed Grand Mosque - A very recently built mosque. We drove past it under construction all the time.
Zakher Lake - Artificial waste water lake. Good sunset and barbeque spot. Do not swim! The first time my Dad and I tried to get there, Google Maps took us right up to the entrance of a palace, guarded by someone with an AK-47. I think that was the palace of a classmate of mine, who was a royal (he had H.H. on the electronic school register). Don't worry, you can reach Zakher Lake normally.
Murabba Fort - Out of place, being in the middle of the Town Centre. Unless the fort used to be at the edge of an oasis and it shrunk.
Segia and Mezyad Forts - Not sure if Mezyad Fort is visitable. If you're considering these, then you probably live in Al Ain and are bored. XD