After going through it too, all the above is very good advice. Except not bothering

... I enjoyed every minute of it, after they had a re-think, the end results!!! My Vulture is just parked up for now (it feels like I'm cheating).
My approach:
Get them open for business first.
If you're into combat scoop everything that ships drop on destruction. Everything.
Go into every Threat Level 0 USS you come across. Scoop it all. Especially look out for the "High Grade Emissions" and "Combat Aftermath" these give the best goodies.
Spend an hour or two planetside, location or planet/moon composition unimportant at this stage. Get used to the SRV and scanner. Pick up everything. You don't need to find any Polonium now so can breath easy... Maybe hit a base or five... Medium security + or ++ avoid those with the large outer tresspass zones (not worth the effort, or a Wanted status) scan all of the Data Points: find them all, then scan them, it's on a countdown timer that resets. They give all the Firmware/Data needed. Modified Embedded Firmware etc. I now have a little list of six places I can hit for all my Firmware needs. They take a week I think to reset.
Get a Wake Scanner and scan wakes at a Nav Beacon or station. I used to hunt at the Beacon of my home system, pick up all the ship drops and scan all the wakes. Just tidying up all the mess I'd made as I went

You don't notice you're ship scanning, although targetting ships in supercruise is a no brainer. You'll end up discarding LOTS of ship scan data, it clogs up the inventory quickly. Watch out for when it's full, you won't pick up any more data but you won't get any subsequent msgs telling you you're not.
Just play for a bit and forget you're collecting stuff along the way... Maybe make a list of the Very Rare stuff so if some pops up as a mission reward you won't miss it. Exquisite Focus Crystals spring to mind.
You don't NEED to hoard the Commodities as many feel the need to. Just needs some common sense. All my ships are completely engineered and I didn't hoard a single item... But if I ever see Micro Weave Cooling Hoses (Dirty Drives G3) or Nanobreakers (Power Distributor G3 upgrades) I will take the mission and do some further rolls on anything not yet to my satisfaction.
Then start targeting the specific upgrades you want. You'll find you already have lots of bits and pieces by now and can target search for what you don't have.
Top of most people's list to go for first is the FSD range upgrade, makes sense, first upgrade I give to all my ships. A Vulture with a 20ly jump is nice to fly places... Available from Felicity or Elvira, both to level five... This is the first real choice; do you want G3 Dirty Drives? (yes, yes you do

). Or G3 Shields... Most opt for Felicity BUT she requires Meta-Alloys to unlock, Elvira needs Soontil Relics which can be bought.
By then you'll be well on your way.
Hope there's something here that helps. My best advice is do it alongside what you'd be doing anyway, unless it's something targetted. It's not a grind or chore then.
Good luck Cmdr
(I feel very compelled to add... G5 Overcharged Multicannons.... The glory... The devastation... The insane grin as you open fire the first time

...)