It depends what kind of plants you're looking at. Personally, I think a couple of small variety anubias and some java fern (perhaps even the narrow variety if you could find it) with some moss would be a good start if you aren't going to be adding CO2 or ferts to your tank.
But if you're wanting plants to help with the cycle, faster growing stems like Elodea and Cabomba are good at sucking up nutrients, but they may show signs of deficiency after a while if no other nutrients (trace elements, macros etc.) are being added.
If you want to properly have a tank cycled before the fish goes in, that's going to take around 4 - 6 weeks and mean you will have to provide a source of ammonia to establish your biological filter and the beneficial bacteria in it. If you want to do a fish-IN cycle, you can add your betta and start performing large water changes every couple of days to keep the water params. in check.
You will need a filter to cycle a tank, so if your tank has no filter, it cannot be cycled, since the beneficial bacteria grow in the filter media. Also under no circumstances wash your filter media in tap water, or change the entire media over, it's just a ploy by filter manufacturers, and it will cause your cycle to crash.
Hope that helps, I know it's a lot to take in