Hi, I've started an online business recently and can share some basic info.
The place you go for registration is your canton's Handelregister.
There are two types of companies, broadly - Personengesellschaft like sole trader or simple partnership, where the owners are liable with their personal assets, or Corporations such as Gmbh or AG where the owners are liable only up to the limit of the company capital.
You can register as a sole trader (Einzelfirma) in which case you're personally liable for everything and your company tax return is part of your personal tax return. A Gmbh (a limited company) will have a potentially higher cost since you need to show CHF 20,000 as capital.
In Zurich the cost for registering a Kollektivgesellschaft (simple partnership, as we're two partners) was CHF250. That was pretty much everything.
Once you're a registered company, you need to decide whether you need to register for VAT (MwSt). If you're importing or exporting stuff you may find it very necessary or helpful.
You'll probably need a bank account but that's very easy to get in a country built for banking
we're at Postfinance who are very start up friendly.
Then if you need a payment gateway for your website (if you're actually selling stuff online) then use Paypal or Stripe as they're very reliable and simple to set up. Work with your website developer to set it up.
This are a few basic steps, have a look at gruenden.ch for plenty of start up info and tips.
I am sure it sounds like a mountain of stuff to tick off, but don't despair, simply deal with them one by one.