Hello Inycepoo, welcome to the forum. I don't have time to go deeply into this, but a couple of quick points:
1) Spending money on an older bike doesn't do anything positive for the value. Put $600 into a $200 bike, and you still have a $200 bike. Upgrade whatever you want if you think it will improve your enjoyment, just don't think you are adding value, or will get that back if you want to sell it.
2) There are almost no* part upgrades that will improve your speed measurably. Better parts are often lighter, and sometimes will be more durable.
3) Lighter doesn't mean much. Compared to the weight of you+bike, shaving a lb or two off your bike is only .5 to 1%. That translates to perhaps .5 or 1% faster on a long climb. You would do better to get a good nights sleep, or eat a better breakfast. I ride a 24lb steel bike, and can out climb dozens of people in my club who ride 15lb carbon bikes. There are even more who can out climb me, probably riding almost anything.
*4) Tires and tubes are the exception to rule
#2 . The best, lightest, most flexible tires can probably shave as much as 10 watts per tire off something like your Gatorskins. You will feel and notice something like this.
If you want to go faster, it really isn't about the bike. Unless something is wrong with your bike, I doubt a brand new, $10k bike would improve your speeds by more than a % or two. (Please don't listen to bike shop salesmen. Their job is to convince you that your old bike is junk, and buying this new super bike will make you instantly faster.)
The unfortunate truth is that to go faster, you need to ride more, and train harder.
By all means, upgrade anything broken, then look carefully at getting the best tires and tubes you can. (Latex tubes are faster, but come with some downsides.) Also, fit is extremely important. Spend whatever it takes, and mess around with different bars and saddles until you are truly comfortable. Getting your fit dialed in is a long, and sometimes continuous process, but it can really help both your endurance, as well as enjoyment.