Rather than fly Hamburg to London and then a long train journey - like 5 to 6 hours to Tenby, it may be more convenient & cheaper to fly from Berlin to Bristol with Easyjet. It looks like they have flights starting on 26 June with Tuesdays, Wednesdays, Thursdays & Sundays being the days of operation. From Bristol Airport, I would suggest that you take the shuttle bus into the downtown station (Temple Meads) and then take a train to Cardiff - trains every 30 minutes & it takes just under an hour. (Cheaper to purchase an Anytime Return). Stay a night or 2 in Cardiff and then take a train to Tenby - which, even from Cardiff, is about a 3 hour journey.
Alternatively, go to Amsterdam Airport which has direct flights by KLM to Cardiff Airport. If you land at Cardiff Airport at an inconvenient time, I suggest you book into the Holiday Inn Express at the airport for 1 night. From Cardiff Airport, a shuttle T9 bus does go into Cardiff but as you are heading west, you do not need to go into Cardiff - which is east of the airport. Instead, your easiest option is to take the NAT shuttle bus to nearby Rhoose railway station (RIA). Trains are only 1 per hour. Go to the platform on the seaward side and a train will come to take you to the main line at Bridgend. From here, some trains go direct to Tenby but the odds are that you would have to change trains en route - probably at Swansea. www.nationalrail.co.uk will have the train fares.
Please note that the Tenby branch line has only 1 train every 2 hours and rail services can be poor on Sundays. From London & Reading - a through express runs all the way to Tenby on summer Saturdays only. (Only the rear 5 coaches will go all the way).
You may also find cheaper flights from Amsterdam to Bristol using Easyjet.
The Pembrokeshire coast is spectacular and I particularly like the sections west of Tenby such as the Stackpole area / Barafundle Bay.
Milford Haven is an oil port and Pembroke Dock is not attractive to say the least.