I rode the Oregon Coast last summer, traveling by bus from the Portland train station to Astoria. Portland train station was very helpful with information. The bicycle and panniers had to be separated, but all stowed neatly under the bus. The bus delivered me rather late (after dark) and I had to ride a bit to my first campground stop at Ft Stevens SP
http://www.oregonstateparks.org/park_179.php
This is a great park, they even had a nice outdoor large screen slide show. I would recommend riding there, even if you don't plan to camp -- especially if the shipwreck is visible.
I was camping, so my bike was fully loaded and *too* heavy - live and learn. I rode as far south as Neskowin and in a town just south of there, there is a bus over the ridge east to Corvalis, then rode to Eugene and farther south. I took the scenic route, including the detour out to Cape Meares; if you skip these places, you may miss some of the most beautiful scenery along the coast. Steep grades? Yes. Worth it? Yes. Remember to wear bright colors and to have rain gear -- even if it isn't raining, the morning fog is dripping!

The train back to Portland would be my personal choice for round trip. Take note: at some point along this north/south corridor, Amtrak requires bicycles to be "boxed". I'd recommend checking with them to find out where that occurs (because, north of that point, you can just roll your bike on board -- much nicer!)

Car rental (one way) is very expensive (from my experience last summer); it was prohibitive, in fact.