Please recommend me bitcoin wallet which can allow me to set really low transaction fees.
IIRC, electrum's minimal fee is 0.000005 (~0.6 cent). Using this fee, even with a very small transaction, this is only 2-3 sat/byte... Using such a low fee will give you allmost no chance of being added to a block, and a lot of nodes will reject your transaction due to to low fee.
If you want an even lower fee (or no fee at all), you can even manually create and sign a raw transaction using bitcoin-cli or the debug window of bitcoin-qt. If you do this, chances of being added to a block are even lower than with the minimal 500 satoshi's of electrum, and even more nodes will reject your transaction.
Remember: fees are not something you pay to the creator of the wallet (except in some very rare cases like old multibit HD wallets). You pay a fee per byte of transaction data to persuade a miner to put your transaction into the block he's currently mining on, instead of chosing a different transaction.
The higher the fee, the more chance the miners will favour your transactions... Since 0 fee or very low fee transactions have allmost no chance of being added to a block, a lot of nodes just reject them.