I have a Nikon S1 that I wanna use the lens from on a D3300. Is there a adapter I can use?
No. That is not really possible without additional optical elements. The S1 has a much shorter flange distance than the D3300, so there are adapters to use F-mount lenses on a 1-mount camera but not vice-versa. Also note that the field-of-view will change completely when you adapt between different sensor sizes, so your everyday lens may not be one anymore. As a matter, lenses for the S1 are unlikely to cover the image circle needed for an APS-C DSLR.
As for something for the D3300, the concept of everyday or walk-around lens is highly subjective. A good amount of people like a wide-to-mid-range zoom but some are OK with a single prime lens and others feel the need for a super-zoom. You will have to decide based on your own experience. Its is much about how you see the world and conditions you shoot in.
Among Nikon lenses, I would opt for an AF-S 17-55mm F/2.8G as a general purpose lens but would actually prefer a Sigma 18-35mm F/1.8. A lighter weight option is the Nikkor AF-S DX 16-80mm F/2.8-4E which seems interesting but I have not tried to yet, so I have no idea how it performs. Again, this is a personal opinion based on my shooting habits.