Firstly, you have posted a query on Wales in the England forum. (RS does not have a UK forum).
It could well be cheaper to fly into Dublin and then travel by train to Rosslare for a ferry to Fishguard (Stena Line) or Pembroke Dock (Irish Ferries). Check air fares on www.skyscanner.net and select “monthly View’.
You are right to also consider Manchester & London Airports. From Manchester, direct trains run every other hour to Milford Haven but it is a very long trip. (MH is the nearest rail station to your start point at Sandy Haven). Likewise, from London to MH is also a long trip. If doing either of these routes, I would consider staying 2 or 3 nights in Cardiff. (Go to the Wales forum where you should find some info on past posts about Cardiff). Here is a map of the rail system in Wales:> http://www.projectmapping.co.uk/Reviews/Resources/ATW%20Route%20Map%20-%20Dec%202013.pdf
Another way of doing it is to fly into Dublin; then fly to Cardiff Airport. Now, you could go into Cardiff but if you want to get to Milford Haven (or Tenby) the quickest & cheapest way (just over 3 hours) - take the shuttle bus to Rhoose station = £1; then catch a train to Bridgend from where you can get trains to Pembrokeshire. (I just check for Today & fares are about £21 but if I were to pre-book for tomorrow - some off peak trains are only £10).
Now, if you were to come on the ferry into Pembroke Dock, you could consider spending time before your main trek in south Pembrokeshire. You could take a train or bus to Tenby = lovely seaside resort. Tenby can also be reached by train from the east.
If you just want to hike the coast, consider taking a bus from Pembroke to Bosherston. Start walking the path behind the church along boardwalks by lily pools. When you reach the coast - turn east. See:>https://www.nationaltrust.org.uk/stackpole
http://www.tenbyvisitorguide.co.uk
http://www.visitpembrokeshire.com/attractions-events/castles-and-heritage/ (See Pembroke Castle)?
Regarding the end of your main trek at St.Dogmaels > you will no doubt go to Cardigan. From here, you could take a bus back east to Carmarthen where you can catch a train to wherever. Alternatively, you could take a bus north up the *coast to Aberystwyth - from where trains go east to Shrewsbury - where you can change for a train to Manchester or continue all the way through to Birmingham International Airport. (Shrewsbury is worth a look - maybe an overnight). https://www.visitshrewsbury.co.uk
However, I would also consider staying a couple of nights in Aberystwyth and taking :>http://www.rheidolrailway.co.uk
Between Aberystwyth & Shrewsbury is Welshpool - the town is not up to much but 1 mile W is:>https://www.nationaltrust.org.uk/powis-castle-and-garden
- The bus will call at New Quay & Aberaeron. You may like consider stopping in one or both of these places for an hour or so & then taking the next bus north - check bus timetable. Aberaeron has a lovely harbour surrounded by pastel coloured houses:>
http://www.aberaeron-wales.com
For Buses:> https://www.traveline.cymru
For trains:> www.natioanlrail.co.uk (Cheaper to pre-book long journeys about 11 weeks out for specific trains - try and book with the company that operates the train).
