I need to buy quite a few bitcoins and was wondering what's a safe and cheap way to do it? Without having to pay any high fees and without the risk of getting scammed.
You can, for example, use one of these: Bitfinex, Kraken, BitStamp, BTC-e These are services with good reputation over a long period of time. BTC-e is somehow controversial, however, as it is operating relatively anonymously in the eastern Europe region. Each of these markets provides different options on how to add funds, so feel free to check them out and pick the one that suites you best.
The fees are different on each site, I do not remember exact numbers for each one, but you will usually pay 0.1 % to 0.25 %. That is a trading fee. There might be withdrawal fees too, depending on the type of withdrawal. For example, Bitstamp had very low fees when using SEPA (bank wire transfer of EUR within EU). In general, if you are in Europe and working with EUR, you should be fine, paying almost nothing. BTC-e has withdrawal fees between 1 % (e.g. for OKpay), to 7 % (Paypal). PerfectMoney is for 3.5 % there ... As for deposits - this also varies, BTC-e has for example 2 % deposit fee on OKpay. 1 % for MoneyPolo etc. Again, using SEPA is usually cheap for deposits, sometimes even free of charge.
I've used localbitcoins and I've never had a problem, it's a bit of a hassle, but the rates are better. It's worth it if you're buying larger amounts imo.