The universe is shared between solo, open and private group modes, so literally the only major difference between the open/private group modes and solo mode is that there is a possibility of you seeing other human commanders out there. That is literally the only significant difference. You can flit between modes as you please and the only other thing which changes between switches is, as far as I know, the mission/passenger board, i.e. switching modes builds a different list of missions, though they will usually still be of the same types, just a different list. This is the basis for "board flipping" to stack multiple missions, which for the likes of me is the only time I would ever dip into open to do anything. If you want to wing up then yeah you have to be in open or private group.
Private group is either something you generally seek out to join, or create yourself with your mates. A popular one is Mobius, which is basically a very large group of players who agree that non consensual PVP is forbidden. In open there are no rules, anyone can grief/gank you if they want. Certain systems and community goal areas are basically grief/gank hubs. You have to be either mental, or really, really keen on the idea of being prepared for pvp combat at all times to play a community goal permanently in open. I myself want to insulate myself from griefers and gankers as much as possible, I am in one of the Mobius groups.
Sometimes I forget I am still in open and I have been interdicted a few times, though luckily it has been a role playing pirate rather than a ganker, and was was actually fairly enjoyable, but the risk of mindless griefing, and the way I know it would make me feel, means I would never play in open for anything other the occasional sortie if I ever get into mild PVP myself, which even that is a way off for me at the mo. Once you have more cash than you will ever need then the game would probably only be fun if you played in open, as by that stage you will probably be well burnt out on the normal stuff.
