Understand the basics of Carnatic music. While traditional Western music features 12 individual notes per octave, in Carnatic music these notes are referred to as sruthis, and there are 22 of them per octave. In addition, the melodies of Carnatic music often slide from note to note in a glissando fashion rather by jumping abruptly from one pitch to the next. As you've probably already realized, this sort of music is impossible to play on a piano with standard tuning.
Purchase a synthesizer. The particular model you choose is not important as long as it has a pitch bender. This is a small wheel, typically found to the left of the keys. The wheel allows you to increase or decrease the pitch of a note in small increments.
Familiarize yourself with any other features of the synthesizer that might further facilitate Carnatic music. More expensive keyboards include a variety of playing modes, such as portamento. This allows the keyboard to fill in the space between each note with a slide automatically. On some keyboards, the speed at which this glide takes place can be changed. This allows you to play Carnatic music accurately.
Use the pitch bender wheel primarily for shaking notes, rather than slides, as long as you have portamento turned on. Carnatic instrumentalists often "shake" a note, or rapidly vary its pitch, while playing. You can recreate this effect easily with your pitch wheel by quickly rolling it up and down. The motion should be subtle, and should provide an added layer of depth rather than change the tuning of the piece.
Consider purchasing a synthesizer with programmable pitch tables. This feature allows you to change the tuning of each key by an exact value, typically in hertz. By programming the notes at your leisure, you can recreate the 22-note scale used in traditional Carnatic music.