Exchanges don't charge fees for deposits. They charge them for withdrawals. If you want to avoid them you can use services like localbitcoin but rates will be a bit worse.
Don't use those scammers at Localbitcoins. They charge ripoff fees way more than it costs to send BTC. Over $20 in fees to withdraw from this ripoff site. Never will use them again.The whole point is they are not scamming but providing an escrow service. Its probably not the cheapest for very small amounts as the fee is not a percentage ? however they have split fee between deposits and withdrawals to even it out and they also vary the fee according to the network itself so it might be cheaper at some times and days then others.
One site I did used to use has never varyed their withdrawal fee and right now its one tenth of bitcoin to move the money, imagine how stupid or dishonest that system is. Localbitcoins is quite reasonable in comparison and I cant agree with much criticism of them from my experience over the last 4 years.
Of course there might be better sites, competition is always good.
I know for small trading the best place might be Reddit where you can switch to various shopping cards and vouchers and people up reputation and so on
