I've stayed at Lake Como a few times now, including a couple of extended stays and even honeymooned there. It's a fantastic place to spend some time. One of the most beautiful spots on earth.
As others have said, you don't want a car, it's more of a hindrance than a help.
The places to stay are the villages clustered around the mid-lake area: Menaggio, Bellagio, Varenna, Cadenabbia.
These are all interconnected by frequent ferry services. We usually get a day pass or multi day for the ferries and visit multiple villages from a base of either Varenna or Bellagio. So yes, stay in one mid-lake village and visit the others using the ferries.
I've also stayed in Como itself, which unlike those above is a small city. I'd only suggest staying in Como if you had things to do in Milan (about 45 min train ride).
A particular accomodation I can recommend, and sounds like it suits your budget is:
Welcome on Residence L'Ulivo Bellagio Website, Holiday Apartments in Bellagio (Italy), 13 apartments for 2 or 4 person Benvenuti sul sito del Residence L'Ulivo di Bellagio, appartamenti vacanza a Bellagio
The story here is that the very expensive 5* Hotel Serbelloni, also has some self contained apartments on their grounds, about a 2 minute walk away from the main Hotel building. Staying here gets you access to all the facilities of the hotel, but at a fraction of the price. About 120Euro per night in October.
You get to use the pool, gardens, concierge services, and restaurants of the Serbelloni, plus you have your own kitchenette, so you can pick up some groceries and a bottle of wine from the deli in the village and cook yourself some nights.
