Your weapon loadout sounds pretty good actually, probably an issue of not timing it right. The starting laser on Kestrel A (Burst Laser MkII) is the best laser and probably the best weapon overall in the game, and combos well with a Flak and a bomb.
With such a setup you would probably bomb the shields, then fire the Flak to weaken them further and snipe the system of your choice with the laser a second after the flak pellets' impact (usually shields, sometimes weapons if they have heavy weaponry you need to disable immediatel). That should be enough to easily dispatch three shield bubbles and even kill four bubbles with a bit of patience.
Your problem probably lies in a weak defense, so that your plan can be easily foiled by disabled systems and you soon begin to struggle. Also, some battles are simply not worth fighting through, mostly because of heavy rocketry oder powerful synced lasers. It's nice to have a backup plan for those encounters or they can be VERY costly or simply lose you the game.
Upgraded engines are very good in such cases, as they let your FTL drive charge considerably faster and at the same time mitigate the damage intake while trying to get away. If you have shit engines, you are forced to sit through the encounter and take massive damage. Cloaking is another very useful system, you can dodge the first missile salvo with a well timed cloak, disable their weapon recharge for a while and let your FTL charge while being hidden. Hacking can buy you time as well and is a very powerful offensive tool in both boarding and gunship playstyles.
The key to victory is often to explore as many beacons as humanly possible. A good escape mechanism lets you abuse this even further. For example you can explore 2-3 nodes more than you would be able to otherwise and then cheese your way through the rebel fleet to the exit beacon with cloak, considerably increasing your resource intake. Especially in the harder modes, you need to grab every resource you can.
Also, choose your jump choices wisely. It's probably pretty useless to visit a shop at the first sector because you couldn't afford any good stuff there anyway and it would waste your precious jump. Rather avoid the shop if you are not searching for something specific (a weapon, a certain system, fuel, repairs) and instead take some nodes that have a chance to gain you supplies.
It is often beneficial to choose routes through nebulas because you can jump to a lot more nodes in nebulas as the fleet is delayed (very much in non-nebula sectors, a bit in full-nebula-sectors). They have a lower encounter chance as well but more jumps usually pay off in the end.
Also choose your sectors to jump to wisely. Homeworlds often have high value quests that yield powerful rewards if you can solve them (and unlock other ships), so taking the homeworld node is worth it most of the time. For example, Zoltan homeworlds are especially valuable because they have a very high reward quest that doesn't involve any fighting at all (only careful choice of dialogue).
Green sectors usually have more stores and more benign encounters, if you are struggling or searching for something specific, the green sector is probably the better choice. If you have a well equipped ship and you are looking for trouble (aka combat rewards), choose the red ones. Mantis sectors feature a lot of boarding encounters, so you should not choose them if you are not equipped to repel boarders. Rock sectors often feature missile weapon enemies, so if you have no way of deflecting missiles it could get rather costly. Rebel controlled and pirate sectors feature a lot of generic combat encounters. The rebel stronghold has a "secret" miniboss fight that can be pretty rewarding and fun if you have a decent ship.
Engi A has about everything you need to get to sector 3 without a single usable drop and comes equipped with some of the most useful drones and weapons right away. Defense drone MkI is the best defensive drone in the game (low energy cost, fights specifically missiles), laser drones are generally more useful than beam drones because they help you to actually push through and the upgraded Ion Blast can take down up to 3 bubbles of shields on it's own (with patience, 2 bubbles pretty effortlessly). It's actually one of the most forgiving ships in the gaming. The biggest weakness is probably being boarded by a lot of scumbags, but there are cheeses to live through most of it.
When you are being boarded with no good crew, gather them in the medbay and vent the rest of the ship. The boarders will take damage while trying to get to the last oxygen filled room in the ship and then have to fight your guys in the medbay, making even heavily outnumbered encounters survivable. When you struggle with boarders, consider getting a timely door system upgrade. It helps tremendously with delaying boarders and softening them up with low oxygen before they reach you.