Does Fees for transactions depend upon wallets we use? Are transferring funds from one wallet to another cheaper in one wallet than another?
Yes, since every wallet has different developers and different ideas. Most wallet that has the same host like coinbase has no fees when sending from coinbase user to coinbase. But there are some wallet which has suggest with a specific which is not totally recommended. And there are some walle that you can input manualy for the fee.
Thanks everyone.
So now as everyone suggested, wallets differ in their fees.
Which desktop wallet is the cheapest according to you? Most of you only talked about exchanges, but what about desktop wallets?
I think you got the wrong idea. The thing is miner fee is the most important factor that determines how fast will a transaction confirms as miners prioritize transactions based on their included fees
and certainly not on what wallet the transaction comes. So if you want the cheapest fee, you can have a wallet where you can edit miner fees manually and set it to 0. That is the cheapest although the odds are your transaction/s will get stucked for a few days and might never confirmed.
To answer why some wallet offer cheap fees but don't get their transaction stucked. Most wallets that does this are those who don't give your private keys as you are not actually the one who sends your coins but them. A good example is what TryNinja stated:
Every onchain transaction will require you to pay the same amount in fees. What will change between two different wallet softwares is the way how they calculate the "recommended fee".
Coinbase to Coinbase transactions are totally different since you are not doing any Bitcoin transaction, but only changing numbers inside Coinbase database. Every Bitcoin you deposit at Coinbase will give you a IOU that is worth what you deposited, and this is what you give to someone when you send them a transaction inside the Coinbase system. When you withdraw, they will remove those IOU from your account and send you real Bitcoins.