Help this Bonsai amateur!!

[Saa May]

Hello everyone! I got a bonsai tree as a gift couple of months ago and since then I haven't done much besides watering it, and even that I'm not sure if I'm doing right, since the lady at the shop told my cousin that you should pour some water into the container in which the pot is placed and not water it from the top?? I'm not sure if that's the right way to do it. I'll attach couple of photos of my tree so you can help me identify it, as well as the pot it is currently in. I'm not sure if I should re-pot it now or not, something tells me I should...The tree itself seems healthy to me, it even has some new leaves and branches which makes me super happy! Anyways, could you please tell me what my first steps should be? Should I re-pot it into something bigger? Should I cut the leaves so they stay smaller, because I'd like them to be that way, and if so, which ones? I personally like the 'upright formal' style and I was wondering if it is possible to achieve that with the tree I have? When and how should I wire the branches? How do I water it? How do I cut it? What do I do with the roots?? H E L P me...I really feel like I shoud do something for the poor thing and I grew to like it very much...Her name is Saa May, by the way Thank you!

 

[Shibui]

I believe that tree is called pot belly fig - Ficus Microcarpa Retusa.
Most ficus are tropical or sub-tropical so make sure it does not get cold. They grow happily indoors.
Many trees die quickly in the hands of beginners so you have done well keeping her alive so far.

There are lots of ways to water. The important thing is to make sure the roots have some water but it does not really matter whether the water comes from below or above. It is important that tree roots are not always wet. Constant wet can cause problems so we must learn to let the treed get a little bit dry before adding water but not so dry that it dies. This is a balancing act. Fortunately ficus are very tough and can survive short periods of dry that would kill other species. I guess your cousin was told to water the pot below as this is a good way to make sure the roots do get properly watered. Sprinkling from the top may not always wet the roots right through until you really know how much and how often to water.
Ficus can also survive for many years in very small pots. Normally bonsai are put into attractive ceramic pots to make them look better. The size and shape chosen to show off your tree as best as possible. Choosing pots is beyond this short post so you will have to make your own decision.
Ficus growth habit is for the first leaf on each shoot to be smaller. Each new leaf gets a bit larger. There are several ways to have small leaves on ficus bonsai. 1. tip prune regularly - allow shoots to grow 3 or 4 new leaves then cut the shoot back to the 1st or 2nd leaf. Do that as often as required. After tips are cut new shoots will grow from the base of remaining leaves below the cut. New shoots will start with smaller leaves and your tree will also have more branches. 2. Selective defoliation - snip off individual large leaves. just snip through the leaf stem of any larger leaf. Trees usually respond by growing a new shoot from the base of the cut leaf stem. New shoots have smaller leaves but additionally the tree will have more branches.
Styling: This tree does not really look like formal upright. You could try to achieve that style but it is difficult to do. Maybe better to just try to make an attractive shape without calling it any particular style for now. Changing the shape can be done by pruning (cut off unwanted and untidy parts and leave the good parts) and/or by wire and bend the branches if needed.

What do I do with the roots??

I'm not sure what you are asking. There are many things to do with roots. When repotting you can cut a lot of the old soil and roots off before giving it fresh potting soil. New roots will grow where old roots were cut and the tree will grow healthy. repotting should be done every 2-6 years in most cases.
If there are ugly roots showing some can be cut off. Trees do not need all the roots they have so removing a couple will not kill the tree and it will grow more in the soil to replace any you have taken.
If you do not like the spaces between the roots the tree can be planted a bit deeper in the next pot so those spaces don't show.