First thing I do when starting a new game is to see if there's any system bounties I can claim in my starting system (or nearby). There usually is.
Depending on what kind of start I picked, I may not be able to fight pirates on my own. In that case I tend to pick a patrol fleet to hang around and join any fights it gets into.
This is also a good time to expand my fleet with useful ships recovered from battle, or just found.
I also make an effort to salavage every wreck and debris field I can. Not only can you cover some or all of your supply costs, it's also a good source of weapons which neutral markets lack.
Recovering Shepherds is one of the highlights of this stage. Sheps rule early game.
Once the bounty has ended, I make sure my fleet is in order and equipped as far as my ability allows.
Then I use the planet list to look for any worlds in the core with ruins, and visit them in whatever order feels most convenient. The salvage from ruins will frequently yield useful things to keep, and various things to sell.
Blueprints are learned as I find them, if I want them. Unwanteds and duplicates are then sold to the pirates to expand my loot selection.
Once all the ruins in the core have been cleared, I will take a few transport missions for the extra money. This also gives me the opportunity to look over many markets for interesting ships etc.
All the time I'm doing this, I'm still jumping on any wrecks and fields to salvage.
I will repeat this until I feel like it's time to do something else. Nothing really set, just when I've 'had enough'.
Now I move on to named bounties. I'll pick one which is fairly close and complete it, then use that as a 'jumping off' point to explore the surrounding area. Once my cargo is full or fuel/crew is low I will head back to the core and sell off whatever I don't want to keep, and see if there's any way I can expand my fleet. Repeat for as long as 'feels' right.
Slowly I will replace my ships with larger ones to allow me to fight larger fleets and haul more loot.
All the while I'm doing this, I am looking for interesting systems which might be suitable for building a colony in.
At first I only bother surveying worlds if they have ruins on them, but eventually I just roll through system after system surveying everything.
At this stage I can keep my fleet out exploring almost indefinitely, as recovered supplies and fuel can cover your useage with the right preparation. The only real limits are crew loss, and cargo space/fleet limit.
If I find somewhere I want to build, I shift my focus to that and then re-start exploring once adequate defences are in position and carry on from there.
If I explore the whole sector and have not found anywhere I would consider suitable for building, I just re-start.