AK,
A few thoughts on your travel plans.....
Your trip appears to be only a very short one week, and it that's the case you have too many destinations listed for such a short time frame (IMHO). Keep in mind that you'll have to allow for travel times between locations, and the usual "benchmark" is to allow about half a day each time you change places. That takes into account not only the actual travel time, but also checking in & out of hotels, waiting for transportation, etc. My suggestion would be to either increase your time or decrease the number of places you want to visit.
It would help to know where you're travelling from? If you're from North America, you'll typically arrive in Europe the day after you depart, so the first day is used for the flight there and the last day will be required for the flight home. You'll also be jet lagged for a day or two after arrival, so won't be up to full "touring speed".
One thing you might consider to maximize your sightseeing, is to travel directly from Zürich to Lucerne, and spend at least two nights there. There's a rail station right at the Zürich airport and the trip is only about an hour, so very easy. I was just there about a week ago and while it's a "busy" place, it's not hard to find the station or the SBB ticket office.
As it's your first trip to Europe, I'd highly recommend reading Europe Through The Back Door prior to your trip, as it has a lot of good information on "how" to travel well in Europe. Be sure to study the "rail skills" part so that you're up to speed on things like changing trains, etc.
I agree with Tom that April is not the best time to travel in Switzerland, so it's good to hear that you're going to move this to June.
and want to explore as much as possible in a decent budget.
One caveat to mention on the topic of "budgets" - Switzerland is expensive, so be sure to allow for that in your budget.
Given the short time you'll be in Paris, I'd suggest packing along a copy of one of the Guidebooks so that you can plan your touring in the most efficient way possible. The Pocket Paris book would be a good one to consider. There's a LOT to see there, and you're not going to be able to see it all on one visit. If you're planning to visit the Louvre, don't try to see all of it as it's HUGE. Most people focus on the Denon Wing and the Mona Lisa. Try not to visit on a Monday, as it's usually packed! Depending on what sights you want to visit, a Paris Museum Pass would probably be a good idea, perhaps the two day version.
Especially in Paris, you'll need to be vigilant to avoid pickpockets and scammers so a Money Belt would be a good idea.
Good luck with your planning.