just to clear things out, if a user made a transaction and broadcasted it using its own node, then it will be sent to other nodes of the network for verification right?
Any node is connected to some other nodes.
Once you broadcast your transaction, it's relayed to some other nodes and finally almost all nodes (including mining pools' nodes) will have your transaction in their mempool.
and when we broadcast our transaction, we pay the fee? and to whom? or we save a fee?
You pay the fee to miners, so that they include your transaction in the blockchain.
Not that it doesn't matter whether you are owning your node or you are using a SPV wallet. You have to pay the transaction fee.
And another question is, if someone has our address then he can still use that address on explorers to check our data right? then what type of privacy does it provide us?
Yes, anyone has your address can check all your transactions.
Assume that you don't run your own node and you are using a SPV wallet like electrum.
Whenever you open your wallet, electrum sends all the generated addresses to a server. The server can know that all those addresses are owned by the same person.
If you connect to the server with your real IP address, the server can link all those addresses to your IP address.