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Trading for Mew is the only legitimate way to obtain a Mew in FireRed anymore. Mew is an Event Pokémon, and was only available during a single event in 2006. There is a glitch that people claim will work to find Mew, but this only works in the original Pokémon games. It does not work in Fire Red. Mew is exceedingly rare, and your friend won't likely give it up easily. You'll need to sweeten the pot, so ensure that you have a good selection and be prepared to part with multiple top-tier Pokémon to get it.  Any of the Legendaries make for good trade stock. This includes Articuno, Zapdos, Moltres, Raikou, Entei, and Suicune. Also, if you have some of the event Legendaries, such as Lugia and Ho-oh, these will make your trade much easier. Mewtwo will most likely need to be traded in order for your Mew trade to happen.  EV-train your trade stock. A group of Pokémon that have been expertly EV-trained will command a much better trade value than freshly-caught Pokémon. Invest some time to build up your team before you start the trade. When you've finally convinced your friend to make the trade, connect your systems and get it done! Walk away knowing that you have one of the rarest Pokémon in the game, and that you got it legitimately.
Find a friend with Mew. Build up good trade stock. Make the trade.