Some of the other answers may provide you with some challenges.
Most important is that Wrangell and Ketchikan are only accessible by air or by Alaska Marine Hwy System. Either way you are looking at Prince Rupert in BC (or Seattle itself) as your jump-off.
Amtrak from Seattle to Vancouver is a smart choice. It is 933 more miles to Prince Rupert, so you may not wish to rent a car.
You can take BC Rail Vancouver/Prince George and connect to VIA on to Prince Rupert, OR you can Greyhound it.
Once in Rupert you can suss out small private-but-scheduled air carriers, OR you could skip BC and fly Air Alaska from Seattle, or you can take the Alaska State Ferry either from Seattle/Bellingham or from Rupert -- I have taken it often and have always enjoyed it.
Based on the Winter schedules, you can have a nice week -- 5.25 hours to Ketchikan, 3 nts there, then 6 hours to Wrangell, 3 nts there and the return from Wrangell to Rupert, 15 hours overnight. Total fare is $169 USD though you may wish to add a cabin for the final leg. If you are thinking of summer, the schedule is much more frequent but book it NOW and be sure to lock hotel space.
DO NOT take a bus to Whitehorse (where I live) -- we'd love to see you, but we're about a thousand miles north of where you want to go -- and in any event, as Badger correctly said, AlaskaDirect does not serve Southeast AK. And there is no rail service between Vancouver and Anchorage, not yet anyway (we're workin' on it!)
You have to understand that 'The PanHandle' is a long way from the 'rest of Alaska' and there is not much at all in way of connection between the two.
That all said -- these two AK communities are hard to reach but WELL worth the time and effort (consider Sitka as another worthwhile jewel while you're there).
Good luck with your travels!