Missions & Complexes

No group does Scout sites. They're not worth the time nor effort and don't help one acclimatise to the proper isk making sites.

For Vanguard sites most groups use the Scimi exclusively, Assaults they'll have 2-3 scimi & 1-2 Basi , and Headquarters they'll have around 5 Scimi, 2-3Basi. So overall you should have better odds getting in as a Scimitar. Depending on the group too, TVP generally cherry pick people based on the fit so you can often expect long wait time with no guarantee of fleet.

Better off training for the Vindi, a group like WarpToMe will accept one with meta4 guns and Vindi are quite sought after by all groups.

Pretty sure you an do them solo, The DPS is not very high.

Scout sites are terrible (although the eve uni wiki is useless and doesn't go wave by wave doing the 1-3 x spawn and then a list)
Intuitively it seems you have to kill the same EHP worth of targets in scout sites as vanguards.
Scout sites may have gotten a payout buff but ccp also put in more sansha and made the mara paleo actually do something (besides being hard to hit it's like a scimitar repping stuff)

For vanguards Scimitar tends to be most popular (but also requires logistics 5 and cap stable and whatnot because it's more important.) because of the remote tracking computers.
Assaults are 20 man fleets that are hard to get right and if you have blingy fits, a good FC and light spawns you might get similar isk/h as Vanguards. (again it's down to EHP you have to grind down and it's just much more compared to Vanguards.) I only know 1 or 2 people who could FC and run assaults decently apparently OCF = good isk/h; NCS = bad; NCN = Lolwut? split up the fleet youdrunkm8?

as for which battleship i'd recommend training into turret battleships as soon as possible because missiles don't apply well or do much damage in incursions because the NPCs have low sig, high speed, and missiles have delayed/wasted damage. (fire missile, thing dies before you hit it, etc.) Even a pancake does badly in incursions (and those missiles go 30km/s; still take 2 ticks minimum to reach NPCs because launch is 1 tick; second tick is hitting.)

An optimal fleet in headquarters has 3 basilisks and 6 or so scimitars.
Waitlists might be longer for logis sometimes because people think it's easy train easy fly and weekends are usually busier as well. Sometimes there's too many snipers, too many vindicators, etc. It helps to be flexible sometimes.
And last off keep an open mind who you fly with. There's propaganda out there mudslinging ya know but what matters is that you get out there and make money. What's important is that the community you fly with makes you the money you want and has a decent atmosphere.
So try out all the things and see what you like :)

("Thou shalt not subject thy community to market forces" is the commandment some communities preach if you fly with others but all i can think of right now is TDF who bans people for flying with other armor communities. (and they may not even do that anymore.))