Welcome to My Swiss Alps!
Rockoyster has mentioned the most scenic route for you, and also given you a link to the timetable. The time table selects the fastest rout unless you put in "via" points to force it to a particular route. The fastest route to Interlaken will be to go from Geneva to Bern, thence to Interlaken. Not as scenic. Faster. Your choice. You don't have al ot of time and that night influence your choice.
Check out the timetable. It will be very very useful.
Make sure to read the instructions on the first page of that link.
This is good for general information:
Rockoyster has mentioned the key point.
It will be late Winter at higher altitudes. Definitely at 2000 meters and above, probably at 1500 meters more or less, the trails will still be blocked with snow. or, they may be unsafe due to the possibility of avalanches.
The floor of the Lauterbrunnen Valley at around 800 meters will likely be free of snow, or what is left will not block trails. Due the location, it does not get a lot of sun.
Altitude is critical. Geneva will definitely be experiencing Spring as will Luzern ( the local German spelling, pronounced essentially the same as the English Lucerne).
This is a good map for general traveling, to go along with the timetable that Rockoyster mentioned. Check out the menus under Points of Interest. Under "Traffic" you can turn o a layer with icons for the transport stations. Unfortunately, it does not show altitudes.
Zoom in to zoom level 32 and you can see the train lines.
Here is a link to a map that does show altitudes. It is dense with information, so it is a bit more complex. Actually, a lot more complex. But, it has altitudes. You can turn on Hiking Trails in the Menu:
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,false&E=2 642035.73&N=1160787.17 &zoom=4
It seems that you may be coming to Switzerland for the first time? If so, the many links and sublinks on this page will be very helpful for general orientation:
For a start -
Harder Kulm -
Harder Kulm will be open. (the German word Kulm is related to the Englihword culmination, and means the top of the mountain, in this usage.
The Eiger Trail -
The Eiger Trail typically opens in late June. Because of snow conditions it may be later in the year. There is a fair amount of water flow from melting snow and ice until later in the year. A significant part of the trail is scree, meaning good boots or rugged hiking shoes, as well as hiking staffs, are necessary.As a result, the trail is maintained frequently, and may be closed on some days in late June and early July. It traverses the lower part of the North Face (Nordwand) of the Eiger. North means less sun than other exposures.
The trail begins at Station Eigergletscher and descends 400 meters at the end, to Alpiglen. If you have not done much hiking on steep hills, your knees will be stressed. Tired., and sore for a few days.
If you walk all the way down to Grindelwald Grund , your descent is from 2320 to 934 meters. Unless you are in good shape, particularly, your legs, that is sufficiently downhill to be very tiring and very very hard on your knees.
Will require serious walking:
You have a steep 400 meter climb over about 2- to 2 1/2 km. from the upper station of the cableway to the "Triftbrücke." In late May, I'd expect snow on the trail, if it is open.
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,false&E=2 669396.00&N=1172432.88 &zoom=7
Open all year, easy to get to from Lauterbrunnen
Oeschinensee (Oeschinen Lake)
At about 1500 meters, the trail may be snowy or slippery, but should be passable. The ski season is probably over at that altitude, so I don't know if the trail is maintained in late May.
There are lots of nice walks around places such as Luzern.
Some excursion transport will be available in the Jungfrau region. The cograil train runs all year. You can go up to Schilthorn. The cableway from Wengen up to Männlichen will open on May 26th for the season. The trail from Männlichen to Kleine Scheidegg rarely opens before June 15th.