This is the best answer and aside from that, you can copy the private keys manually then export it in your new wallet
And if you are going to continue using Bitcoin Core, then it's better to copy default directory so that you don't want to download blocks from the start.
You can find wallet.dat in default data directory. See https://en.bitcoin.it/wiki/Data_directory.
One thing, so if I manage to download the whole blockchain with bitcoin core as I am in fact doing, if later I take that wallet.dat file from bitcoin core, could I use this file in another lighweight client (electrum) to check my coins?
No, especially electrum allows you to restore its addresses via a mnemonic called seed. You could import the private keys and addresses from your old bitcoin core wallet into electrum, but it would defeat the purpose of the seed. The imported keys are not covered by the backup via seed. If you use a different wallet the most safe to do so is to transfer all coins from your old wallet into the new one. If you cant get bitcoin core to work for some reason for this, ask someone you trust to do it for you or import the private keys into a temp wallet.
A wallet is not the same as an address, by default bitcoin core only manages a single wallet file the wallet.dat. You can rename the file to e.g. wallet1.dat and bitcoin core would create a new wallet with a new file and new addresses for you, but there is no way to manager several wallets within a single wallet file in bitcoin core. You would have to close bitcoin core and rename the files every time.
Thanks, got mixed up. Bitcoin core only manages one single wallet file at a time, noted. (It's with armory where I have more than one).