Things to do in Zurich and Murren that aren't enormously expensive?

Zurich can be expensive! True! So, try to avoid the traps and get a ZurichCard. 

Zurich is a very historical city. So there are many interesting things to discover in and around the city (such as the reformation church, etc).But that's what every tourist would do.

But here's something very cool: Zurich in June is fabulous! Pack your Swimsuits and go for a swim in the lake or the river in one of the many public baths. This is something you don't have anywhere else …

• Seebad Enge
• Flussbad Oberer Letten

Talking about the lake – take a 90 minute lake cruise!

'Rent' a bike (i think they are still free) or stroll around the city (means walk!!) … Kreis 1 (Niederdorf / old town), Kreis 4 (Langstrasse / alternative entertainment area), Kreis 5 (alternative and modern) … and dive into the vibe. There are some world class museums (such as Landesmuseum which is a bit like the Swiss version of the Smithsonian) and the ZurichCard gets you in for free.

I know – coming from the US might raise many questions about safety etc. since Americans are generally overly cautious. DO NOT. It's safe - even at night!

Bottomline, hopp on a free bicycle and explore, go iE to Frau Gerolds Garten, get a Beer & Bratwurst and sit with the crowd and chat with the people you meet (very international), let your instincts guide you and not as expensive as you hear, if you go where the locals go and do not eat in proper sit down restaurants (these are in fact very expensive), go to one of the hundreds take aways (asia, india, pizza, proper burgers, etc.) and eat there. Avoid chains you know (Starbucks & Co.) and if you need bottled water, buy at the supermarket (not the newsstand) – otherwise drink tab water.

A few things to do in Zurich that are cheap:

1. Go walking on the Uetliberg, the big mountain to the west. There are trails all over and the view from the top is lovely. There are tree carvings in the forest. You can make this more expensive by eating in the restaurants but you can also grab a cheap sausage + frites at the top.
2. Walk along the lakeshore and up Bahnhofstrasse. If the weather is decent enough a quick sausage and beer on the lake front and just enjoy the views.
3. Take a train to Lucerne. Trains are relatively cheap and Lucerne is lovely.

We took the boat tour around Lake Zurich and it was very affordable.  We also did a trip to Felsenegg for a memorable dinner.  It was a short train ride and quick tram ride up the mountain - had a blast  http://felsenegg.com/

I'd take the gondola (or walk downhill) from Murren down to Gimmelwald to visit a quaint Swiss mountain village. Then on to the Schilthorn if its clear on top to check out the view. Rick Steves writes extensively about the Lanterbrunnen Valley and the Berner Oberlandat his website.