Thank you for the explanation!
I'm just now starting to look into bitcoin technicalities and therefore I had a few doubts.
How do you check the current backlog of transactions waiting to be confirmed?
A lot of people use the
https://bitcoinfees.21.co/ website to get a pretty good look into the current state of transactions and fees.
Others have written their own software to monitor the memory pool and recent blocks of their own full node.
And how do you do those type of calculations into knowing what kind of fee should be sent to hurry the transaction up ?
A well written bitcoin wallet should already be doing these calculations for you and recommending a fee to get fast confirmation. Unfortunately there aren't very many well written wallets yet. I've found that the Bitcoin Core wallet "Recommended" fee has worked well for me as long as I've had it running for at least 30 minutes (preferably longer) before I send my transaction and I set the "Confirmation Time" slider all the way to the "fast" setting.
It needs to be running for a bit so it can collect information about the transactions in recent blocks as well as transactions currently unconfirmed. If you just start it up and immediately send a transaction, it won't be able to calculate accurately.
If you know which outputs are going to be spent by the transaction (if your wallet provides coin control features), and you know how many new outputs you are going to create, then you can get a pretty good estimate of the size of the transaction before you send it.