Bitcoin Core Problem To start things off I am new at this so bear with me...
I have bought coins through LocalBitCoins and sent them to my wallet (with 24 confirmations).

I am currently on Bitcoin Core with 2 years and 23 weeks behind Synchronizing. I heard you were able to send Bitcoins without actually being Synchronizing but whenever I try to send my funds elsewhere I get a "Insufficient funds exceeded" box. I go back to overview to see that my Bitcoins have not been transferred to my Bitcoin Core wallet yet probably prompting the insufficient funds.

I know waiting to finish Synchronizing is the better solution but I thought I was able to send the coins without being synchronized?
As far as I know, you heard wrong.  I don't think Bitcoin Core supports sending coins when the wallet is not synchronized.  It needs an up-to-date copy of all the transactions on the network, so that it can be sure of spending coins that have not already been spent.  (Such a transaction would be rejected by the network, but it would lead to confusion on your end, and could also result in other nodes on the network refusing to relay your other legitimate transactions.)

You can either wait to synchronize (note that you are downloading about 60 GB of data) or see Cannot access my bitcoins because my wallet is still synchronising. What can I do to spend them immediately?