Enter and register under your Swiss passport. You won't gain anything with the Greek one and might wind up with more hassles/paperwork. It's almost always cheaper and easier to be a local. Yes, you need to register at the municipality. Not sure on cost. I paid CHF 104 for my first permit, if that helps.

For insurances, as Medea says start with Comparis. To save money, take the mandatory basic insurance with the highest franchise (deductible). Don't bother with supplemental coverage. Depending on the model you choose (standard, primary doctor, Telmed, etc.) expect to pay 200-300/month based on your age and gender. If you can afford to pay 6 months up front, you might get a discount.

Share a flat to save on expenses. You'll still want some sort of renter's insurance. If you're not in a remote area, use public transport instead of a car and you'll save the maintenance/insurance on that. Invest in a half-tax card or monthly travel pass for the area you live in. The half-tax card is around 150 annually. Monthly travel passes vary depending on region. Liability insurance isn't that expensive, and is highly recommended.

If you're just earning odds-and-ends with tutoring, you won't be paying much income tax. You will be paying VAT (8%) on your purchases, so try to be thrifty with spending. Shop at Aldi and Lidl for basics, and if you need to shop at the big three (Coop, Migros, Denner) try to catch things on sale or even marked 50% off at the end of the day.

Keep in mind - with many things, cheapest isn't always best. So do your homework. Start with what you need and want, then choose the least expensive option among those that meet your criteria.