I think the Premier Inn Roath is in a fairly unappealing retail / business park type area, but it's a short walk to Newport Road from where there are lots of buses to the centre. To walk all the way to the centre is probably around 25 minutes or so.
Definitely recommend the castle. Pay the full £11 for the guided tour - it's well worth it and necessary if you want to get the most out of the full craziness of the Victorian decor. Also the new interpretation centre will give you a good background. Reckon on 2 hours at least to give the castle a good visit.
Depending on what you're into, the open air National History museum at St Fagan's can be a pretty fantastic place.
Full of reconstructed Welsh buildings from the last several hundred years, transplanted from across the country. Also plenty of ancient trades on show and you can buy a lot of the stuff they make.
The other 'must see' museum is the National Museum and Gallery of Wales in Cathays Park - slap bang in the civic centre adjacent to the castle. The art galleries contain one of the best impressionist collections outside Paris. You should have a walk around the rest of the civic centre complex itself as well - there are a lot of very attractive buildings and the War Memorial / Alexandra Gardens at the centre of the complex are beautiful.
You should definitely try Cardiff Bay if you have the time. Pop into the Millennium Centre (again, tours are available and worth it if you like theatres - it's pretty impressive). Also have a wander round the new Assembly chamber designed by Richard Rogers. The bay (Mermaid Quay especially) is also good for an evening out. It's more focused around restaurants than bars so tends to have a more laid back atmosphere than the city centre after dark. There are a smattering of good bars / pubs down there though. Bar Cwtch is delightfully cosy and independent-feeling, The Waterguard is smart, Salt loud, Eli Jenkins traditional, so there's a decent variety.
Also the Cardiff Bay barrage is a terrific piece of engineering and actually a very pleasant place to be as much of it is landscaped. You can either get a boat across the bay to the barrage from Mermaid Quay, or simply walk across it (takes about 20 minutes). If you like that sort of thing, it's fun watching boats locking in and out of the bay through the barrage locks - when the tide is out the difference in water level between one side of the lock and the other can be around 10 metres.
Otherwise if the weather's nice a walk through Bute Park (behind the castle) is a very pleasant way to spend an afternoon. If you have an hour or two to spare you can walk through the park along the river to Llandaff Cathedral, which is small but architecturally pretty interesting. The cathedral green / Llandaff high street are pretty, and you can bus it back into the city centre easily from there.
Hope that's a start!