That is not the easiest area to do without a car.
There is public transport but it isn't comprehensive and relying on it will limit the areas you will be able to visit.
You will find details of the local bus network here
Train information can be found here
Is there anyway you could hire a car? It will make your trip much easier when you are in Wales and the driving is no where near as bad as it sometimes appears during discussions on this site :-)
I can't provide any advice for a local guide but I know I have seen some recommendations on this forum before so it is worth doing a search. You are allowing yourself more than enough time to visit the locations you have listed. You could certainly fit more in.
Snowdonia is beautiful and the castles at Conwy, Caernarfon and Beaumaris are worth a visit.
Portmeirion, Bodnant Gardens and Plas Newydd are all nice trips if you like gardens and country houses.
The small town of Llangollen is pretty and has some nice shops. Mold is not pretty but has a good traditional market!
Chester is a good base to take the train to Liverpool which is definitely worth a visit. Port Sunlight is also an interesting "village"easily visited by train.
As a tourist town Chester has a good range of hotels and B and Bs. Those in the town centre tend to be more expensive and car parking is in short supply.
Two slightly more suburban areas worth looking at are Hoole and Hough Green/Curzon Park.
Hoole is a Victorian suburb that has become quite cool (coole?) in recent years, but not unbearably so. It has a nice high street with some good pubs and restaurants. It also has Seabreeze the best fish and chip shop I have ever visited. It's one of the few things my brother and I agree on,! :-)
Hough Green is on the Welsh side of town so has good links over the border.
If you can try and stay somewhere on the same side of the city as your relatives, if you planning to visit them regularly. There is a lot of building work and road works going on over the next few months and it is causing havoc with the traffic ( according to the daily phone updates I get from my mum!)