Welcome ricoh83! Great that you get to spend your honeymoon in Switzerland. Since you're traveling outside of high season, you won't find the most common tourist attractions to be that crowded. So visiting highlights such as the Jungfraujoch and Schilthorn near Interlaken
and Rigi, Pilatus and Titlis near Lucerne
is an option. Both towns are perfect in terms of transportation.
However, if you prefer to stay away from such areas, I recommend that you take a look at Graubünden
which is generally more quiet and at least as beautiful. One of my favourite regions, the Engadine, isn't well suited for a stay in April though, as it's high up in the mountains and snow may still be melting. But you can consider areas near Chur, Davos, Klosters (these have the best public transportation connections), Sedrun, Bergün/Filisur, and so.
April is a good month in my opinion. You won't be able to make hikes up in the mountains, but spring nature is beautiful. A disadvantage is that quite some cable cars and mountain trains won't be running yet. As for weather: I've had quite some beautiful April months, but there are really no guarantees.
Tips for cheap accommodation and other budget tips are listed here
If you plan to travel by train and bus mostly, a Swiss rail pass
would definitely be worth your money. However, I'd wait with ordering if I were you; new promotions for next April may come along in 2013. If so, they will be mentioned in our rail forum
. You can start planning your trips with the aid of the timetable
. Have fun preparing!