I usually use potting soil in the house as it's sterilized.* Garden soil can be a bit heavy in pots and will definately have 'critters'.* You might want to consider taking cuttings and rooting them to bring indoors.* Then plant in potting soil with some perlite added.* Be sure to not breathe in the dust from the perlite as it's bad for your lungs.* You can fertilize with an organic fertilizer like fish emulsion or fish emulsion mixed with seaweed or sea kelp to add nutrients.* The plants will be slowing down for the winter so you only need to fertilize once or twice all winter.
Another trick I use with my house plants, especially those that bloom, is to use milk.* When the container is empty I add water to fill and use that to water my plants.* I also use cooled water from boiling eggs as they appreciate the calcium.* Cooled veggie water is great too if no salt has been added.* If I steam veggies, I cool the water and then fill the cooking pot with more water and give it to my house plants.* I've been doing this for years and never have problems with them.* My peace lilies bloom alot from these methods.
As to the kind of pots to use, that would depend on how you water.* If you are a person who tends to water alot, then clay pots would be good as they breathe and tend to dry out faster.* If you forget to water until your plants wilt a bit, then plastic pots could be an option.* Overall, I prefer clay even though I tend to let my plants dry out before watering.* Since the begonias don't need to be constantly wet, the clay might work for you too.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.