June is a 'long' month in Switzerland, so far as riding the passes is concerned. As others have mentioned, you'll likely find most passes still closed (snowbound) at the beginning of June. If I recall correctly, the Stelvio is one of the higher passes, I very much doubt that it will be open during the first half of June. But, having said that, I think that the Stelvio is over-rated so far as riding pleasure is concerned.

Marc-S has provided a good list of some of the more enjoyable passes. My personal favourite is the Sustenpass.

Although you might be travelling alone from the UK to Switzerland and back, you won't be alone once you are in CH... you will find hundreds of riders at the various cafes near each end of every pass.

Some suggestions:

1) Buy your vignette (a toll sticker) at a gas station near the border before you enter Switzerland. Cost is about 40 CHF.

2) Whatever else you do, be certain to respect the speed limits within towns and villages. The Swiss cops are pretty reasonable, and tend to leave bikers alone out in the passes, on the understanding that they will strictly respect the speed limits within the villages. If you hoon your way through the villages, you will be shunned by others at the cafes, and will go broke paying the photo-radar tickets.

3) Get a GPS that has the capability to warn you of photo-radar locations, and be sure you have an up-to-date database of photo-radar locations in it. By 'up-to-date', I mean no more than a week or two old.

4) Switzerland is not cheap, primarily because the Swiss Franc is very strong against other currencies at the moment. If you want to save money, stay at the youth hostels - they welcome people of all ages, and are very clean and well run.

Michael