To trade between Emerald and Fire Red, you need to beat the E4 first. Make sure you have at least 60 Pokemon in your Pokedex before beating the E4. Once you beat the E4, Professor Oak will give you the National Dex.
That's not it though. You need to activate the Network Machine in One Island in order to trade with Emerald. Two items are required for it to work and you need the National Dex before you can start this event. They are the Ruby and Sapphire. The Ruby is at the bottom of a cave in Mt. Ember. The cave is guarded by two Team Rocket members. One will give you a password that you will need later. Once you retrieve the Ruby, give it to Celio in One Island so you can start hunting for the Sapphire.
To get the Sapphire, you need to beat Team Rocket in the Ice Cave on Four Island. After that, go to the Ruins Valley in Six Island to find the Sapphire. A scientist will steal it from you and will tell you the other password for the Rocket Warehouse located on Five Island.
Once you are in the warehouse, simply beat everyone in there. Team Rocket will abandon the warehouse and the scientist who stole your Sapphire should still be in there. Defeat him and he will give the Sapphire.
Give the Sapphire to Celio and the Network Machine will work now. You should now be able to trade with R/S/E now.
Yeah, there is a lot of stuff to do before trading with R/S/E. I personally hate doing this quest.
As for battling between Emerald/Fire Red, I'm not too sure about that.
