I'm in the process of building a new computer and have already received some parts and just waiting on funds to purchase the remaining parts. My question is will I be able to just transfer some of the parts from my current computer into the new case and use the new parts with everything working properly? I just feel like my current one goes too slow and it's pretty much unplayable.
Your current processor should still be good, just overclock the crap out of it, if you do a clean windows install on the SSD your computer will fly. The graphics card is due an update though, but there will be new ones in a couple months up to you if you want to wait, but they should be substantially better then current ones.
Bit of a waste that you already bought new memory, the new skylake CPU you intend to buy may be around 20-25% better then your current one but in most games you will not notice the difference as they are mostly GPU limited.