I have been wanting to decorate my living room for awhile now and honestly I just don't know where to begin. It is the room you see when you first come into our house, but we are still in there quite a bit so it needs to be liveable. I want new furniture, a new light fixture(s), carpet, accessories, and wouldn't be opposed to painting,but I'm not sure how to do that without having to paint the whole kitchen too since they share a wall...
You will have to- never stop the paint in the middle of the wall. It's no big deal- just one Saturday :) I will likely have to do it in stages too for financial reasons so I hope you can still help me.
Almost everyone does this kind of thing in stages- the important thing is to plan it all out before you begin- this keeps you from making mistakes. Where do I start? What should I choose first?
I don't know where the focal point should be since there's no fireplace or other obvious focus to the room.
Good job recognizing that you need a focal point. By process of elimination- I would have it be the wall adjacent to the window wall. That means that we need something really "cool" on that wall. 
It isn't a very large space and I have young children so it needs to be family friendly as well as affordable. So, I'm coming to you for advice and inspiration and hopefully enough bravery to actually do something. I guess I'll start by describing the room. The room is 13feet by 14 feet with a banister for the stairs on one wall and a window on the opposite wall. About 4 feet of the longer side is wood floor and really more like the entry way than part of the room because of that.

The "dining area" is also visible from the front door if that makes a difference. I want it to feel welcoming and cozy without feeling small or crowded. I would like to get new furniture and I was thinking of a couch and two chairs unless there is a better combination of furniture you know of for this space. (I'd also love to know the best way to arrange the furniture--the piano has to stay in this room)
I made a floorplan according to your description and you do not have enough room for a sofa unless you put it along the window wall- which I don't recommend- or along the wall that should be your focal point wall- which would be bad- as it would face the entry wall- go with a love seat. (photo below from Pottery Barn)
Keep the scale of your furniture small. In my diagram- I used a 64 x 36 love seat- and two chairs- that are about 32 x 32.
The chair above is from Padma's Plantation- I would use different fabric- here is one from Beacon Hill- This type of chair would also be super easy to re-upholster if little kids give it more wear than you would like.
but the chair is great- it is only 30 x 32. The taller back gives it the visual weight, however to balance the love seat nicely. Sometimes a smaller footprint takes you to chairs that are visually too small. I also like how this chair gives the room texture and style. This is not very large- make sure you do not purchase pieces that are too big.I have looked at the local furniture stores in my area and just haven't been able to find anything I love enough to buy. I scanned a picture of something I got out of a magazine awhile ago, but the store is no longer in business(see attached). I like the colors in this picture and the shape/feel of the couch. (would you consider that picture contemporary?)
A sofa like this can be very transitional in style.- You could find a love seat in a similar style- keep it under 64 inches in width- but also not too small- you want it as close to to 64 inches as you can find.
I've been thinking that I'd like to keep the furniture more neutral so it will be more lasting, but still allow me to change things up with accessories.
A nice safe way to go when you are not seriously confident about what you love. I'm not sure what to call my style because there are so many different things I like. I don't want it to be too trendy, but I do want to love it. I am open to consider any ideas you have for me.I like all kinds of different colors besides those in the picture too,but I don't have any pics of them. How does the crown molding effect what will look good in the room?
Every room looks more polished and finished with crown moulding. My biggest problem is fear that whatever I do is going to look bad so I just don't do anything.
Do what I tell you to- and it will look great! If there's anything else you need to know from me feel free to ask. Sorry this is so rambling...Thanks in advance for your help!