Although this is early season in the area you should be able to do some of the easier hikes then as most snow has passed. That said, some of the higher elevation hikes described below (such as Männlichen to Kleine Scheidegg, First to Bachalpsee, etc) have occasionally closed due to snow even in early June, but usually only for a day or two.
Since "easy/average hiking" is completely subjective depending on one's hiking experience I have only listed below the very easy walks. Should you want more variety please send me a private message with your email address. I've compiled a 35+ page document on the region based on my 17 visits there. It's written from a Wengen base but is very easy to translate to a Grindelwald base. Many more hiking (and other) suggestions in that.
Regarding reversing your itinerary, I don't think it will make that much difference weather-wise. If you are flying back to the US from Zurich you will be much happier going to the Zurich airport from Lucerne (less than an hour) than from Grindelwald (3 hours and usually 3 switches). Departing from the latter makes for a very long travel day.
Okay, onto the easy hiking recommendations. Will apologize in advance if it's difficult to read. This is a cut and paste from the document I mentioned above and often these things don't do well with this site:Männlichen to Kleine Scheidegg.
This is likely the most stunning of the easy hikes in the region. It is about 3 miles long with a gentle downhill descent on a very wide path with spectacular views of the Eiger, Mönch, and Jungfrau. To start the hike from Grindelwald, take the train from Grindelwald Hbf to Grindelwald Grund (the first stop) to connect to the ski gondola to Männlichen. This is a glorious ride as it is one of the longest gondola runs in Europe. Once atop Männlichen stop to admire the views down the east to Grindelwald, to the west to Wengen and the entire Lauterbrunnen Valley. Then catch the trail down to Kleine Scheidegg and be sure to have lunch there. The views are simply stunning. The restaurant right by the train tracks is superb, you can sit outside if it is sunny or inside if not. The salads are spectacular, the wurst and rösti second to none, and don’t miss the Apfelkuchen mit Rahm (apple cake with cream - DO NOT skip the cream). Portions are so large you can easily share (especially the Apfelkuchen, don't try to eat a whole serving yourself). Take the train from Kleine Scheidegg back down to Grindelwald.First to Bachalpsee.
This is another do-not-miss walk. Take the ski gondola from Grindelwald up to First, which is a treat in and of itself. Then head out on the wide, well-marked and well-traveled path to Bachalpsee. If you are lucky you will hear and see marmots along this path. On the return to First you will be treated to amazing views of the Wetterhorn. It will take you about 2 - 3 hours for the entire round trip, but very easy. You can then take the gondola back down to Grindelwald. If you are up for more walking past First, continue on past there to Grosse Scheidegg, about another 1.5 hours. Stunning views from there, have a drink and/or snack at the inn there, then take the very exciting and scenic bus ride back to Grindelwald. Grütschalp to Mürren
. A trip to the village of Mürren is highly recommended and the walk there from the gondola station at Grütschalp is breathtaking. Approximately 3miles on flat terrain, this walk provides a spectacular view of the Eigermassif, but from a different perspective than the Männlichen to Kleine Scheidegg walk. Tobegin, take the train over to Lauterbrunnen, then cross the street (oruse the underground walk) to the gondola to Grütschalp/Mürren. The trail to Mürren beginsat the Grütschalp station where you depart the gondola. See next hike description for another way to do this walk.Lauterbrunnen to Stechelberg.
This is another must-do hike as it provides aglorious trip down the entire Lauterbrunnen Valley. The valley is full of waterfalls – the literature claims 72 but, depending on the time of year, youmay see as few as 10. Take the train from Grindelwald over to Lauterbrunnen, exitleft from the train station, walk through town, and directly onto the wide footpath that runs down the Valley. The path is paved about half way then turnsinto a wide and well-graded dirt and gravel path. You will pass right by the famous Staubbach Falls. Then after about a total of about 30 to 45 minutes you will see a cut off (to the left) that leads across the Valley to Trümmelbach Falls (see some good pictures at http://graphics.stanford.edu/~lucasp
).This is well worth doing -- these are falls that run through the center of the mountain. Very secure footing and railings let you get an up-close look. After touring there, you can either walk on the road in front of the falls down to Stechelberg (continuing south) or, preferably, cut back across the meadow and rejoin the footpath. Stechelberg Inn lies at the end of the Valley and is agreat lunch spot. The entire distance, including the jog over to the Falls, is about 4 miles on flat terrain. From Stechelberg you can take the bus back to Lauterbrunnen or, even better, walk back to the Stechelberg Schilthorn gondola station (quite close) and take the gondola up to Gimmelwald then on to Mürren. After exploring the village there you can either take the train back down the hill to Lauterbrunnen or do the nice walk over to Grütschalp if you've not done that yet.Lauterbrunnen or Zweilütschinen to Wilderswil, optional to Interlaken
. This is a lovely walk that parallels the river from Lauterbrunnen to Wilderswil and also covers some beautiful forested areas. This path is in two stages, both approximately 3 miles inlength and on flat terrain. The first stage, Lauterbrunnen to Zweilütschinen,starts right at the Lauterbrunnen train station. Follow the signs to the Parking Garage and WCs east of the station and you’ll come upon the trail signsdirection Zweilütschinen and Wilderswil. At the very beginning, there is a short descent over a path that can be a bit muddy and a wee bit steep but quickly puts you out onto a very wide, well-maintained Wanderweg. This stage of the trail follows directly along the river through a lovely forest; you will also see a fish ladder about half way on.
If you prefer to start in Zweilütschinen the trail continues on to Wilderswil. The local train companies have done visitors yetanother favor by constructing a lovely “Nature und Bahn” (loosely translated:“Nature and Railway”) path on this stretch. Cross the train tracks at the northend of the Zweilütschinen station and look for the signs direction Wilderswil and Nature und Bahn trail. This stage is also about 3 miles (flat) and passes through more forested area and the charming village of Gsteigwiler. Admire the beautiful woodwork on the homes and the lovely gardens in town, then continue on the Nature und Bahn trail (Note: there are a number of other paths that also lead to Wilderswil, but the Nature und Bahn trail is the most easily navigated. It is well marked, just look for thespecial signs.)
. At the end of the walk, you will cross over the river via a covered bridge and enter the town of Wilderswil. There are many good restaurant choices in this area if you’re ready for lunch. Some are before you cross the bridge, another good choice is the restaurant right by the train station after you cross over. The village of Wilderswil is another half mile or so west of the train station and also has very good restaurants. You can catch thetrain at Wilderswil back to Grindelwald.
The Nature und Bahn trail ends in Wilderswil but, if you’re ready for more, pick up the trail to Interlaken across the street from the train station in Wilderswil. A short walk of 45 minutes will put you in town, where you can browse the shops, stop for lunch, and catch the train back to Grindelwald.The Planetenweg
. A gorgeous walk along Lake Brienz provides a different perspective than the mountain walks. Take the train from Grindelwald to Interlaken Ost. Walk west fromthe train station and cross the river over the first bridge you come to (just before the Hotel Beau Rivage). Pick up the Planetenweg trail here and follow it as long as you like. You will pass through a number of villages: Ringgenberg, Niederried, Oberried, and ultimately Brienz. Each village has good lunch stops and train, bus or – better yet – boat access back to Interlaken then back to Grindelwald.