Best way to get from Prague to Croatia's Dalmation Coast.

Can anyone suggest a nice itinerary where I can visit Prague, get to Croatia, explore the seaside islands. I can take three weeks to do this trip. What months would the best time be to take this trip? Along with that, some suggested places to stay along the way. And options on how I can travel from start to finish, where I can experience the countryside, the cities and still make my way through it all. Places I should not miss, since traveling in that part of the world will be my first.

I know it's a lot I'm requesting with all my questions... so gladly will take what I can get. Lastly, will be on a modest budget, hostel or small inns work. So thanks for the suggestions. Oh, I'm starting my trip from San Francisco! Many thanks!

If you're on a tight budget, then I'd definitely suggest pre- or post- (high) tourist season (lower prices) - downsides may be less frequent transportation along the coast and to- and on the islands, and not-so-stable weather conditions.
But, it really depends on whether you want warm sea and tourist crowds around you anywhere you go, or you'd like more leisurely and quiet time? Are you more into, say... hiking and strolling through nature, or looking at history and art, or... something else?

I'll try to compile a list of places to visit, and I'll also try to find some affordable accomodation options, but it'd be much easier if I knew the answer to above question. Thanks.

Oh, one other thing. When you say you have three weeks for travel - do you mean three weeks for Prague AND Croatia, only Croatia, or total for the entire trip? Just to be clear on that.

A great way to do your suggested itinery (although it's not for me to tell you which route to take, to the exact inch), is a train from Praha to Budapesta, two wonderful cities. In Budapesta - get a spectacular view of Buda and Pesta across the Danube. It flows through more countries (10) and capital cities (4), than any other river in the world. From Budapesta there is an excellent rail service to Athena - via Lake Balaton (one huge stretch of water, some 80kms long), Slovenia, and into Croatia. The train stops in Zagreb for about 30 minutes. From Zagreb there are trains to all over Croatia.
I would suggest (only my opinion) that you do the trip in late May/early June. The trains are not the best - and the air-con doesn't always work properly. In July and August, that region of Europe can be as hot as Mexico!!
I did the Budapesta - Athena train. It took over two days and was remakably cheap. I paid around (I think) €50 one way. My route back was different.
Have a great trip - and try to spend 3 or 4 days in Dubrovnik. The Croatia/Bosnia border is just a 20 minute bus ride from Dubrovnik centre.