If you just started a new char or you just tiered an old one, you may want to check with other people where they have been and what they have killed as maybe they were leveling there or farmed for materials so the XP will suck. Yes, we all love the multi (as drops increase too if a mob is multi).

  Start in academy, keep killing there till you have all system slots filled and weps of your liking. Pick up all extra items and sell them to the drydocks bottom side of map, you'll need the cash for new hull/drive/items. When you feel ready to move kill the guardian, get the key card and go out.(probably best time to move is when mobs circle color turns light green - select a mob then press alt key to check circle color)
Once out of academy, take your time to check all starbase shops for upgrades. When you're done, return to the place where you entered starbase and go right to leave it.
Press shift + M to bring up galaxy map then press alt key to see location names. Cythris is where starbase is located.(you can return there using /hyperspace)
You can shift + right click on the map to auto-navigate to a desired location.
You can also ask for a place coords, first 3 letters will suffice.(eg. /ask scy will return the coords of scythe) Use /nav xxxx yyyy to auto-nav, careful tho as most likely will get you in the center of the quad and get you killed. Always watch scanner when auto-naving so you can avoid troubles. Move with keys/mouse to stop auto-nav.
Scythe outside is the next stop, kill them all till again mobs turn light green.
Then head to Tyvorax outside, when done (about lvl5-6) move inside.(look on the scanner for the docking place, looks like a flashing red rectangle, when in position look for the green sparkles under your ship to dock) When you reach the bottom right side of Tyv inside, kill Tyv Master and get the key. Do not go to lvl2 unless you're around lvl7. If so, go in and kill around. When about lvl8-9, /hyperspace to return to SB, then go again to Scythe, this time go inside.
Lvl10-11, go Kraxis, should be easy prey, when done , go Tearix.
Lvl12-14, Kitalpha and/or Algedi, when done you can hit the ruins top right of Algedi (careful of Enraged mobs).
Lvl16-17, Thuban or Menkalinan, when done you can check the other inner roids ruins ( save the one right side of Menkalinan last, mobs lvl20+)
Lvl19-20, Unukalhai or Lesath, kill last ruin mobs too (also you can go back and kill some enraged from older ruins).

At lvl20 you can equip a 3rd weapon , do so,  will help alot killing.

Lvl22-23 Zoldar or Koni, first one you can reach easy from SB, go south then pick left side gate to Medusa. The second , go south and pick south gate to Sirrah.
Lvl25-27 Calarb (Sirrah side) or Arbenteur (Medusa side).
Lvl30-32 Kakarin (Medusa side) or Myrmans (Sirrah side) or Rigel(Sirrah side, this is a bandit base, group recommended).
Lvl34-35 Nidlex (Medusa side) or Alosont (Sirrah side)
Lvl37-38 Cron (Medusa side) or Bartroson (Sirrah side)
Lvl 39-40 Alioth outside (Medusa side) or Izar outside (Talitha) or ruins around Sirrah side.

  At lvl40 you can equip a 4th weapon.

Lvl42-43 you can hit Errai ( bandits Medusa side ) , for all bandit bases it is recommended to be grouped, them hit pretty hard.

  After lvl44 it is recommended to be grouped at all times, soloing gets harder.

Lvl44-45 Izar inside (Talitha) or Mintaka (bandits south from SB)
Lvl46-47 Acamar (Talitha) or Botein (Medusa) , both are bandit bases.
Lvl49-50 Altair (bandits Medusa) or Nogrisurnik (Sirrah) or Alioth inside (Medusa)
Lvl54-55 Markab (bandits Medusa) or Okda (bandits Talitha)
Lvl57-60 Celbalrai (Sirrah), Honru (Medusa) , Hydra outside (Talitha) , Cerebus outside (Medusa), Perceus outside (Sirrah)
Lvl62-64  Zaniah (bandits Talitha), Hydra in , Cerebus in, Perceus in also you can start killing Talitha ruins, start with the one top from Talitha then the one left side from Cetus.
Lvl65+     Kill other ruins from Talitha, inside Crypt (right side of Okda) if you have a TKR with key.
Lvl70+     Hadar (bandits Sirrah) or Nashira, careful both places are very hard.

Thanks to Rowspan for indexing locations.