Heater: I really have no suggestions for a heater... All I can warn you about is the last heater I had (Tetra Whisper 2-15) was ridiculous it was automatically set to heat the water to 78 but couldn't even keep my 3 gallon at a steady 72. I definitely wouldn't recommend that heater. Now I have a Jager Adjustable heater for my new ten gallon, but I think I accidentally destroyed it (I thought I could totally submerse it, BRILLIANT) so now I'm letting it dry hoping it might work, since I never plugged it in it just might.... Though, even if it does work, it might not work too well with my set up, atleast an inch of it has to be out of water, only I need to have my water level very high for my make of filter. I might buy a Fluval heater when I get the money. It's fully submersible, and it's totally reflective so it blends with your tank.
Live Plants: I've heard some easy, basic plants are Anubias Nana, Java Fern, and Swords. Most people say Amazon swords, but I also like Ruffled swords, the edges of the plant leaves look all wavy and scalloped... Also Chain sword if you want some shorter plants. Water sprite is also supposed to be easy, and maybe you want a floating plant, like Najas Grass, only I hear it grows huge! and sometimes clutters the tank. And also java moss... I think it looks cool and it offers some nice hiding which makes the betta feel safe and secure!
Bigger Tank: Depends on how big your tank is now... But I'd say anywhere between 5-10 gallons is a GREAT size, and 3 gallons is comfortable but also the minimal size... Whatever size you go with, just don't get a very vertical tank, bettas are lateral swimmers, meaning side to side, so you'd be better off with a rectangular tank, it also gives more surface area, for the bettas to gulp air.
Good luck, hope this was helpful!