Don't forget also that you oinly get Federal Navy (rank up) missions from FRIENDLY federation aligned factions.
No point running around doing missions everywhere, better to stay in one area and focus on repping up a few specific factions.
All federation aligned faction rep missions will help toward the Federation reputation so do any that are easy or in sync with your travels but if you travel too far, you will stop seeing missions from the factions you've already put effort into. I stayed in an area of probably around 30-40 ly and bounced between every outpost and port to find every mission relating to a specific party. I just focused on donations, data and boom supply deliveries. I also always picked the reward that offered most rep rather than a bit of extra cash.
I config'd my Python for cargo (FSD only 15ly) and checked every faction to see if I could double up on some delivery missions. Sometimes you see a trend for certain goods so if you have cargo space to buy a bit more and keep some in store, you might get lucky with certain missions and already have the gear in the hold. If you run out of space you can always sell the stuff you don't need. I bookmarked two or three main 'supply' ports where I would get stuff like food or tech and of course would check their mission boards as well as I knew I would be back there a fair few times. I'd avoid missions that sent me in the other direction from where I was going to avoid straying too far from the start point.
You need four rank-ups from zero to get to petty officer, so you need to have at least four federation aligned minor factions at friendly and above. Donation missions are great and of course, avoid missions with factions that result in damaging your rep in the ones you're building. In a couple of cases I was offered kill missions that were too hard for me in a Python so I just cancelled them and found I was eventually offered another one from another contact elsewhere. I also messed one up so I think I ended up building rep with a few more factions just in case. I was fine doing delivery runs even at elite level although I'm pretty good at avoiding interdiction. Maybe getting an A rated thruster helped. I also always flew with power maxed on engines, the rest on systems.
Final thing that confused me was once I got the mail to say I was promoted and received my Sol permit, I still couldn't plot a path to Sol and saw I hadn't received it yet. Somewherre else here I read you need to get to a station within 15ly of Sol and then it appears, so just set your path to the nearest system you can reach and you will probably find you get your permit by then.
Hope this helps, I add that I'm here thanks to the advice from everyone else online so I take no credit for any of it, just adding the points I didn't see mentioned everywhere.
Edit: By the way, My understanding is that REP builds reputation (popularity) with a faction and INF builds that factions influence, ie the control they have in that area. So my understanding is that you need to focus on REP (that's what gets you to friendly for rank up missions) with high influence or controlling factions to make sure they stay around long enough to help you rank up. shouldn't be an issue at the lower levels as you should be able to reach petty officer in one or two good sessions.