Hiking in early April - which place is best as base?

You're definitely going to want to stay low down unless you like trekking through a lot of mud. One idea thats fun, scenic and easy, is to hike along Lake Luzern on a route marketed as "Weg der Schweiz". I've hiked portions of it (its even ok in bad weather). Pick a section that sounds interesting and take either a steamer (preferred) or train to it. Start hiking and hike as long as you'd like. The views are superb (as are the views from the boat getting to your starting point), there are towns along the way where you can grab a bite to eat (there are also small grocery stores in every town you walk through to get even cheaper food). Its pretty flat and as I recall, parts are even paved or on gravel. Its accessible fairly easily from Zurich or Basel via a relatively short and scenic train ride to Luzern or Brunnen.

http://www.weg-der-schweiz.ch/en/the-swiss-path/en-sections

Montreux has an almost Mediterranean climate, so you might have some good hiking around there.

In Valais, if you stay within the Rhône valley, there's probably some hiking trails that connect the towns together. You shouldn't have any problems here in April.

Also consider the Swiss Jura mountains, which are the lower mountains in the northern part of the country. Easily accessible from Basel and Geneva.

Hiking in the Alps is a very low yield proposition during this time of year.

I'll confirm with my friends over there with first hand experience and boots on the ground, but in many regions of the alps in Switzerland, its been a bad snow year and the paths might be dry at the slightly lower elevations and might be passable. It'll still be brisk up there and the grass might not be emerald green, but the peaks will still be snowcapped and the wild flowers might be out in force. Stay tuned.

Thank you all! I will be going to Lugano and perhaps also Lucerne.