Hi Shreya -
For information about the Luzern vicinity, check the link that I gave you. In addition to the usual tourist destinations for dat trips, there are all kinds of easy or difficult walks around Lake Lucerne, many connecting to the lake boats.
Engelberg is easy.
Interlaken is a stretch, making a long day, especially since you really need to go on to Lauterbrunnen and Wengen for the scenery. But, mid April is not he best time to visit that area. Between seasons; many tourist services are not open.
I am discussing Luzern with someone else and have just posted information about a day trip to the Rigi, with a nice easy very scenic walk, as long as the weather is nice. Look in the last posts in this thread:
There are some discount airlines that have varying schedules for Geneva, Zürich or Basel airport, often from Luton or Stansted.
Zürich is the easiest to use for access to the middle of Switzerland, but Basel is only an hour from Luzern or i hour 13 minutes by frequent trains .......,after you take the shuttle bus from the airport to the downtown main station in Basel. Anywhere from 1 1/2 to 2 hours including the shuttle bus. Z252rich airport to Luzern is an hour twice per hour. One of those trains requires no change at Zürich, the other does.
Geneva has good rail links,including very scenic but slower ones, to Luzern. Fastest go through Bern; 2 hrs 57 minutes. A nice ride with pleasant scenery.
I'd go as late as possible in April.
Check the timetable:
( Caution - the time table shows fares that are already discounted by 1/2, because essentially all Swiss Residents have a "Halb Tax" or "Halb Abo" discount card.
That helps explain why travel is less expensive for the Swiss than for those who pay full fare.)
And, it defaults to fastest route. To force it to a slower route, you need to put in a "connection" or "via." From Geneva to Luzern, that is Zweisimmen, for example.
April is not a busy season in Luzern. Even so, hotels are more expensive there than in the countryside. After all, that is where all the tourists come to spend money....;-)
www.myswissalps.com/pl antrip/doityourself/sa vemoney
I stay in inns in the countryside or small towns. Some have excellent kitchens and good wine cellars, and are much less expensive than the hotels in the cities. Hard to set that up for a first time traveler, though. That is why I suggested Luzern.