When i open serial monitor the green conect led on modul starts to light.
I connect mate to arduino-->RX-TX,TX-RX and i'm powering it with 5V. Do you have any ideas what else to try?
2. I don't know which way to power everything.
USB, batteries, and wall warts are the usual choices. Solar is a bit too limited.
I'm thinking of lilyPad voltage regulator(1.5V AAA batery with step up regulator to 5V). It can provide up to 100mA. Mate needs 50mA and arduino(i guess) 25mA. Powering the circuit for sensors it's aprox. 1mA. I don't know which AAA rechargable battery to choose? I couldn't find the data about capacity of alkaline batteries..
Forget it. You'll burn through batteries in nothing flat. Use 4 AA batteries, instead.