You got already excellent advise here. However it is also important to know your priorities and fitness level. As you come up to the altitude here, I would suggest to take it easy, as you will probably feel it (and then not sleep well the first night).

Your time here coincieds with the start into summer season, so not everything will be open, but things get on the way just this weekend.

Exploring the St. Moritz area for the first half day is a good idea, however if you think St. Moritz is a big place, it is not. If you stroll through the pedestrian area to look for your dinner location, that is almost it. I would plan more time for the very scenic surroundings and nature instead - that is what you come for, not the buildt up area (the typical small villages are better for an impression of local culture than St. Moritz itself). But it is absoluetly good to stay in St. Moritz proper, as this way you are at the public transport hub and don't loose a lot of you precious little time you spend in the region, especially so during still shoulder season. And as Hambagahle mentioned, make sure your accommodation offers the complimentary Engadin pass with two or more nights - see https://www.engadin.ch/en/mountain-railways-inclusive/

So for your first half day, I would suggest you walk to Lej da Staz, a nice stroll along the lake and through light forest to the romantic mountain lake, however the restaurant there opens only on June 13.

If you don't want to hike, you could also opt to go up to Muottas Muragl for a very special dinner experience - actually it's "sunset dinner", however in June the sunset is rather late. The place is just wonderful with panoramic views of the valley, the mountains and glaciers and the lake plateau and the time around sunset is really dramatic. The hotel/restaurant opens on June 8 for the season https://www.muottasmuragl.ch/en/about-us/arrival/operating-hours/

On your second full day I agree with post 1 that you should absolutely experience the Bernina Glacier area https://www.pontresina.ch/en/experiences/world-of-glaciers/bernina-glaciers/. As you mention to be in Zermatt later, I suppose you are travelling on the Glacier Express the following day, so I think you will probably not have a full day of train travel on your day here (also meaning a repeat altitude change twice if you go all the way to Poschiavo and Tirano). I would suggest to take it easy and go to Alp Grüm only and back from there (if you have the pass, this is covered all the way, if you go further on the Bernina line, you would need a return ticket from Alp Grüm on). On either your way there or back you could go up to Diavolezza (train stop Bernina Diavolezza = stop on demand, then up with the cablecar, included in the pass) and enjoy the views and the high mountain feeling while having lunch on a sunny terrace. Coming down continue towards St. Moritz and get off at the stop Morteratsch (stop on demand) for a stroll into the Morteratsch valley towards the glacier - very impressive already after the first bend and becoming increasingly so the closer to the glacier you get. Unfortunately the show cheesemaking dairy opens only a week later, but Morteratsch is a rewarding place anytime. The path is easy and also for inexperienced hikers and there are informations provided along the way about the glacier and the environement.

Thank you so much. I will take your suggestions.

Yes my hotel does provide the free Engadin Card. and indeed I will be taking the Glacier Express to Zermatt after the 2 days.

So you can explore the area without bothering to buy a ticket. Enjoy yourself and take it easy! Spring is slowly arriving here as well and you should find a lot of wildflowers on the alpine meadows.

Thanks again.

Can I check what is the weather like at Alp Grum / Berina Glacier area in June? What clothing should I have on?

June is a transition month, so it can be anything and all - snowfall or warm sunshine or rain or something of all of them. But at this altitude, it fortunately never gets HOT at least. As from where you live you are probably not used to cold temperatures, bring clothes in layers to put on and take off as needed and a waterproof item just in case. If the train has an open carriage, this would be wonderful for the Bernina line, but cold even with bright sunshine, so for this part and also to go up to Diavolezza, bring anything you have to be prepared.