At the start of Prospector, you really need to cherry pick what planets you go for. You don't have the equipment (or the cash) at the start to do much else. Avoid creatures, space-ships and everything else entirely if possible. Sometimes you can risk it for easy minerals on worlds with lifeforms, but it often just means that you'll die early.
Find lifeless planets, explore and grab minerals, repeat. Once you start trading those resources and mapping data you can start to get some decent weapons and armour for your troops. The cash bonuses get pretty massive later on in the game, more than the minerals you're trading. Smith Heavy Industries is my favourite station type, just because you get a bit of extra cash early on and friggen huge bonuses later, for doing what you'd be doing anyway.
The easiest start that I've found is the Troop Transport with 2 medics, 3 squad leaders, the rest as snipers. Upgrade all your officers, with a priority on doctors, then gunners, then science officers (but hire any officer that's higher in level than your current one, keep the next highest one, fire the rest). Special troop types are awesome, as are multiples of officers. Red-shirts might get an extra HP pip at high levels, but squad leaders, medics and snipers ensure you never need it. Early-on your specialists are far better. Ships doctors are very important, and medics stack with them (an extra level worth for every two medics, so bring lots of medics), so it's pretty easy to have almost instant regeneration after your upgrade your ship a couple of times for away team capacity. Squad leaders leading four snipers are great for the massive to-hit bonus early on as well. Get (and level) red-shirts after you've got an elite team going and heaps of excess cash. With the high cost of decent weapons, armour, equipment and ships (for bigger crews) early-on, crew cost isn't a huge concern, quality of the troops using those weapons is. Dead stuff never saps your HP, medics remove the consequence of environmental hazards, plus you'll be punching above your weight at the beginning which cheapens your equipment costs/needs. Which is your main early constraint in Prospector. Simply buying enough plasma rifles, armour, jetpacks and stuff is really damn pricey. Once you have enough, most planets are a walkover and you can level new crew members, buy stuff and get biodata/minerals as and when you want (by wiping out anything on the planet with f 0, or aimed attacks).
Basically there's four phases in Prospector. Beginning, when you've got almost nothing. Early, when you've got a bit and most planets are do-able but challenging. Late, where most planets are walkovers. End-game, where you're on a robot and crystal killing spree in your disruptor laden alien ship, just rounding out your stuff for the final battles.
Get past the beginning phase and the rest just follows on naturally. Unless you're really not paying attention, have a meteor-moment, a brain-fart or a game crash.