If going to Caerphilly Castle (from Birmingham), leave M4 at Junction 28 (on the west side of Newport) & take A468. You might also like to know that immediately south of this junction is Tredegar House.http://www.nationaltrust.org.uk/tredegar-house/
The M4 around Newport has lots of gantries with electronic signs. Be sure to apply any temporary speed limits as they all have cameras on the reverse to catch anybody who exceeds the limits.
Tintern Abbey is just a ruin and I can't see much point in paying to go inside. Likewise with Caerphilly Castle. I have been inside but you can get be far the best views from the outside (for free) - especially from the SW.
Cardiff Castle, on the other hand, does have far more to see on the inside - from the ornate Victorian section to climbing the Norman keep in the centre and looking out across the city. 2 prices are available with the dear one being for the tour with guide (of the Victorian bit). If you have time, I would pay the lower price to go inside on the self guided tour which includes the banqueting hall where Barack Obama attended a banquet during the 2014 NATO summit.
Cathedral Road is about 1 mile long and I presume that the parking for your apartments is accessed from the rear lane. Ask them which side of the road your apartment is located (west or east) and where is the access point for the rear lane? (It is likely that parked cars will be all along Cathedral Road).
I suspect that your apartments are more towards the southern end of Cathedral Road. That being so, it is about a 20 minute walk to Cardiff Castle. Look at a map or Google Earth and find 'Sophia Close' - coming off Cathedral Road. If you walk through Sophia Close and pass the temporary Coach station - turn L and head over the river footbridge & then turn R - you will find yourselves walking through Bute Park. Continue on path passing flower borders - with river just to right. Just after the flower borders end, if you detour to left (east) you will see the west side of Cardiff Castle. Now go back to path and through arch by Pettigrew tearooms & turn left and walk to see main entrance of the Castle. On the opposite side of the road, you will see the Castle Arcade - well worth taking a look inside as it dates from the 1890's. The main shopping area of Cardiff now starts here running south and eastward from where you are now. The City Hall & National Museum of Wales are about a 10 minute walk NE of the Castle.
It takes about 2 hours to drive non stop from Cardiff to Exeter (for Dartmoor). The unique rocks that you find at high points are called Tors = granite.
Also consider doing B'ham > Monmouth A466 down Wye Valley to Tintern - with maybe a detour into Chepstow. (You can view the Castle form the car park &/or walk & view from river bridge). Then, back to A466 for the 1 mile link to the M48 and drive straight to Cardiff accommodation. Dump stuff & walk to Cardiff Castle / city centre or even a visit to Cardiff Bay.
Next day > do Castell Coch first. (On exiting this castle - a back road to Caerphilly is hard left by CC exit - you will turn left at the next junction after the Black Cock Inn to reach Caerphilly via this short cut. (Only if you don't mind narrow roads). Then do Caerphilly Castle before heading to M4 at J28 (Newport). The advantage of doing things this way is that you get more time in Cardiff and are able to take the shorter route over the Severn estuary via the M4 bridge and M49 rather than the longer route from Chepstow via the M48 bridge and the congested M5 passing Bristol. (Look at it on Google Earth to see what I am on about).
This option would leave the evening for perhaps a quick 2 mile drive from Cathedral Road to see Llandaff Cathedral.