First stop trying to do everything all at once, even with 10 starting planets, you can get into trouble very quickly (as you've discovered).
Remember in SotS Prime you started with Destroyer class ships and you build lots of them. Battleriders are the equivalent to Destroyers, but need a "Carrier" to move to other systems. (...and need to be researched as well, both Battlerider tech and carrier tech). This is the "cruiser" age, your ships will cost more.
As A-N has said, prototype first turn every class (it's FREE) to your specifications, but after that be careful about prototyping since that can get expensive fast if you don't have a solid economy and you do it often for every little change. Research Orbital dry docks so you can do weapon (and some modules) upgrades, but you can't change engine or armor on old ships. Just weapons of type (energy to energy and ballistic to ballistic etc) and modules (since they tac onto the ships externally). Do not think of Retrofitting as a full on ship redesign, it isn't. It is only weapons as I've explained and modules. However this keeps your ships from becoming to obsolete and reusable for much longer then before.
Colonize slowly, or use the incentive / Stimulus option and have the AI do it for you (civilian), wait for them to develop the planet for you and once it's reaches a viable age, coax it back into your empire.
Warning using Stimulus takes patience and time, however it can save you a tremendous savings over time and cause you go get big dividends later. SotS 2 is a slower paced more thoughtful game then SotS Prime was. Don't be in a hurry, depending on the map and opponents the game can literally last thousands of turns. If you want a quick bang bang game, play on a much smaller map with less players, otherwise, enjoy the ride!
Remember when you colonize to do pretty much as A-N has suggested. I would make the additional suggestion that if you want to hammer colonization directly that you should research the techs to make it easier for you, like the larger class colony ships. Use multiple colony ships to build them up faster, when (if) a fleet of colony ships are available, but no planets are and you have one in the "oven", send it over to assist the building of the current colony!
Remember to setup what fleets you have to patrol and guard your systems. You don't really need to have one fleet for every system/planet you own, use the mission system!
This is especially effective for Hivers and other FAST movers (like Morrigi fleets with the Gravboat in them etc), or Sol Force etc.
As A-N said, the more ships you have in survey fleets (especially those that carry drones and BR's), the faster you can survey. This is especially useful for LARGE solar systems.
With 10 planet empires at start, you should on average be earning (as in goes in the bank) 1mil a turn, unless you are in some kind of emergency mode. Saving for that rainy day when it's time to revamp the fleet with new prototypes and so on, or the placement of defenses or bases etc. Remember not every system needs a class 5 Naval station and Research and Civilian station, they can get VERY expensive and take a LONG time to build. Speaking of stations etc, remember to use more then one Construction ship, they stack and will let you build those big stations in a reasonable period of time.
Hope that helps along with what others have said. Primarily just remember, this is a thinking players game, slow down, take time, be patient and ....think!
My favorite saying works well for this game...work smarter, not harder.
Good luck, and enjoy!