Tanks should have a 50% water change once a week if they are cycled, with daily changes being necessary with an uncycled tank. All you need to add is a dechlorinator to remove the chlorine from tap water, and most of them also bind metals. Bottled bacteria products are not needed if the tank is cycled.
Despite Fishmanic's nephew having a 1 gallon tank, that is really far too small. My betta's tank is 7 gallons (26 litres) and I am thinking about getting him a bigger tank.
Bettas are tropical fish, they need a heater. Most of the information you need is in the Betta Care Sheet link in NickAu's post. You'll find there that bettas need a temperature of 76 to 82 deg F (24.4 to 27.8 deg C).
The tank will need a filter, but not one with a powerful water flow. Air powered sponge filters are ideal for betta's tanks. But with this type of filter you can't easily steal some media from the platies tank's filter to seed the cycle so you will need to do a complete fishless cycle. If you use a filter which can accommodate media from the platies' tank, the cycle will be much shorter.
Betta's tanks should have quite a lot of plants and other decor. Live plants, particularly floating plants, are better than fake plants. If you choose to use fake plants, silk are better than plastic as plastic plants tend to have rough edges which will cut a betta's fins. Decor should also be smooth. Run an old pair of pantyhose (tights) over any decor and if they snag, so will a betta's fins. Choose another piece of decor, or sand down the rough bits.
Bettas need their own food, pellets made specifically for them. You can't use the food you are currently feeding your platies.
Read through Nick's link on the Betta Care Sheet. It will help you to understand a betta's needs so that you know what you need to buy for him.