The best way to arrive in Llangollen is by car or bus. The A5 (from London to Holyhead) runs through the town. It's worth noting that on busy weekends the car parks within the town centre quickly fill (both short and long term), and the narrow streets slow to a crawl, so during these times, it's advisable to take public transport to access the local attractions: there is also a bus service from Oswestry, about 15 miles away.
By train you can reach either Chirk, or Ruabon, from there check for local bus's to complete the journey into Llangollen.
Llangollen is a small town and all the tourist attractions can be reached on foot. The town centre basically consists of one main street running between the A5 and A539, most of the other roads lead off this. There is cheap car-parking in the town-centre, just off the main street, but note that on busy weekends and holidays, it fills quickly.
As you approach the town you'll see the Pontcysyllte aqueduct carrying the Ellesmere Canal over the River Dee, its over a 100 feet high and a 1000 long. The aqueduct is quite an impressive sight, especially when you consider that it was built over 200 years ago.
There is a ruined castle, Dinas Bran, just outside the town. It is an open site (i.e. no admission charge), and the view is worth the long walk/climb. Just behind the main stretch, on the other side of the canal, is the beginning of the walking path leading up, perpendicular to the canal. There are signs for Offa's Dike, though this too leads to the castle.
Visit Plas Newydd a half-timbered black and white house in its own grounds. The inside has been intricately carved and looks surreal to say the least!
Every year Llangollen hosts an International Eisteddfod where people from all over the world take part in musical and dancing competitions. 
The Llangollen Steam Railway departs from the station in the town. You can enjoy a 3-course meal on the train or just go for the ride.
Horse-drawn canal boats are available 50 meters from the town center at weekends and during the summer. Make sure you sit on the right side of the boat and then you'll be able to look over the side and down a 100ft drop as you pass over the Pontcysyllte aqueduct!
The fast flowing and rocky River Dee is popular with canoeists. You'll have to bring your own canoe though.
The town has the usual selection of gift shops that you'd expect to find in Wales. There is/was an antiques centre.
The town has a wide selection of restaurant types, including Indian, Italian and Chinese.
Gales Wine Bar. It's the business! Second only to The Watergates Bar in Chester!
Most of the pubs are worth a visit. The Bridge End Hotel is popular with bikers, gays and cross dressers and often hosts rock discos at weekends.
There are a number of cheap Bed & Breakfasts in the town.
The Wynnstay, Smithfield and Bridge End Hotels also offer cheap Bed & Breakfast accommodations.
The Chainbridge Hotel and Wild Pheasant Hotel just outside the town are more up-market.
Carry on along the A539 - you'll end up going through the scenic horse-shoe pass, though avoid this if you don't like heights!