There's so much you can do with your fireplace now, the finishes seem to be endless. I would look at the veneers, too, or any finish you want, really. I think real brick might look nice, but your room is so light, am wondering if the paint finish a bit more distressed to reveal some of the undertones of the brick might look nice. But veneers are great, would change your entire room.

First off I'll say I think this space has great potential. Not sure what your style is or what the style of the house is, but it has some midcentury vibes. 1. Take your time with any decisions, if you just purchased / just moved in, no need to change things right away. The reason people always give this advice is it's easy to change your mind and if you haven't really thought about it for long enough you might do something you no longer like that's not easily reversible, like painting brick. 2. Delete wall paneling if possible. If not, paint white like the upper portion of the walls and ceiling. Or better yet lose that dark trim up top. 3. I would lose the curtains but I don't like window treatments, so... 3. Carpet --> light laminate or wood flooring like you said. In the zone of natural white oak. I know you mentioned this and obviously more $$$ and not a bandaid fix, I'm just affirming it's a good idea down the road :) 4. Get rid of shelving on fireplace, and ALL knick knacks that go with it 5. Put a nice looking set of fireplace tools on display 5. If you DO install light wood flooring in the future, consider a mantel that matches instead of the dark rustic looking one that is there. 6. Find a nice piece of art that speaks to you and elevates the room, larger than what's currently in place, and hang it above the mantel. Right now the art fades into the brick. 7. I don't know what is entailed, but if you can get rid of that hearth / semi-circle, get rid of it. Maybe when you do flooring there will be an opportunity to wrap demo of that in. 8. Splurge on a cool modern fan or interesting pendant light fixture to replace existing fan. Maybe add a table lamp to a side table for some additional light. 9. I would not paint or wash the brick, but I don't like painted brick. If you don't like the brick you might as well drywall over it. Again, just my opinion. I think there are a lot of other things you can address first, and then you might find the brick isn't an issue. Just some ideas. When I first moved into my house I also wanted to paint our brick fireplace, white. I'm soooo glad I didn't. I realize everyone has different styles, but just to show you how the surrounding details can make a difference, here is the listing photo and how it is today:

Moving soon and want to update the living room to make it a little more modern. Maybe plaster over brick fireplace, update built-ins, floors, etc.

I suggest you continue with the same flooring throughout, looks like the flooring mentioned is what you have on the right in your dining room. I'm unable to recommend a wall colour judging from your photos but would suggest painting the crown moulding white and not to outline the corners of the wall on the right of the fireplace. The mantel could be black or white, I'm showing it in black because I find it coordinates nicely with the piano you have on the left. Since you have a small budget I didn't suggest removing/replacing anything from the brick fireplace.