I would drop Chichester but add Winchester instead as it is on the way to Dorset (Lyme Regis). You might want to consider doing that by train London (Waterloo) to Winchester and continue by train to Dorset as driving around Southampton and the Bournemouth area can be very busy with traffic. Train to Dorchester South might be a good idea.
However, another possibility which avoids the congested Southampton / Bournemouth area is to go by train (or hire car) to Salisbury - which has a wonderful medieval cathedral. A shuttle bus also goes the few miles up to Stonehenge. (If hiring a car in London, you should drive direct to Stonehenge and then down to Salisbury). So, assuming you have a hire car at Salisbury, it is relatively easy run down the A354 to Dorset . If in Dorset, I highly recommend Lulworth Cove & Durdle Dor.
https://www.dorsetaonb.org.uk/location/lulworth-cove_visitor_centre/
Driving from Lyme Regis up to Bath is not easy and Bath is congested. You will find it much easier to do this bit by direct train from Dorchester West to Bath. Once you have had a couple of fights in Bath, take the train to Cardiff and see the attractions mentioned earlier plus the waterfront (Cardiff Bay - 1 mile south of the city centre). This will take about 2 full days.
After Cardiff, I would hire a car (that can be returned elsewhere) and head west along the M4 to Pembrokeshire (the south-west County of Wales). On the way - just over 1 hour from Cardiff - detour for 2 miles off the A48 expressway to see the National Botanic Garden of Wales. Then continue westward where I suggest that you should make for Tenby in south Pembrokeshire for maybe 2 nights before heading to St.David’s in north Pembrokeshire.
Once done with Pembrokeshire - drive up the west coast of Wales to Snowdonia = the County of Gwynedd.
Once done with Snowdonia - where you will find more info in postings in the Wales section - consider going to Chester for a couple of nights. From Chester you can take a short day trip by train to Liverpool.
If you are going to continue up the western side of the UK - take a detour to see the Lake District (NW England).
Trains > https://www.traintickets.com/?/
Long distance (say over 1 hour) trains are cheaper if pre-booked for specific trains say around 11 weeks out. These are called Advance tickets. If you were just making the short hop from say Chester to Liverpool and back - you would pay on the day. Play around with the above website and see what is says about timings and prices.
https://www.traveline.info is good for finding buses.
Also look at Google Maps for the routes you plan to travel. Zoom down and go to street view to see the type of roads and major junctions. Make sure that accommodation has parking for sections of trip where you will use a car. The Travelodge and Premier Inn chains usually have free parking except in cities. Of course, you may get a better experience by staying with locals but that may not be easy for such short stays.