The coast road is incredibly beautiful, and very twisty and populated by suicidal white-van-men and rust-buckets with no brakes. I can recommend the ride down to Karlobag, then cut up the mountainside over to Gospic if you want a good flavour of that road. Ride defensively and let the other buggers kill themselves. Avoid at the weekend (holiday traffic from the north and from Zagreb).
Places to stay... hmm, beaches are generally rocky, and sand is rare, but Krk on the island of Krk has a nice old town, (I'm told Baska has a really nice beach).
On the mainland, anywhere that grabs your fancy really. Croatia isn't as cheap as you may think, Slovenia is cheaper. In Slovenia, Piran is an old walled pirate town (limited parking, may have to leave your bikes in a carpark on the edge of the town, but I'm not sure...it may well be different if you have a room there)
Up in Slovenia, consider a route via Cerkno/Gorenja Vas/Ziri/Tolmin would be great(my fave part of Slovenia). Cerkno has one big monstrous (but nice enough) hotel and a couple of bars - quaint. Tolmin is bigger. Skofja Loka is a medieval castle town, bit more lively and probably worth a stay. There are hotels/B&B in or near all these places.
Further away, if you are in the area, the road along the Austrian border east of Lavamund is a nice bit.
I've only used the Autozug a couple of times and not on that route, but I never had any problems, other than the cabins being a bit stuffy, and being terrified of falling out of the bunk on every slight curve in the tracks (this will not happen, just in half-sleep the curves seem a lot sharper than they really are
) The have wheel-wedges a bit like in the chunnel train, and also strap the bikes down well. No problems. You'll need to take anything valuable and toothbrush etc with you as you can't get to the bike until your destination. I've found it a good option, just a bit too expensive for the routes I want.