Pick up a kit like this from your local hardware store, it will include every thing you need.
Make sure you measure the room and make sure the last row is not going to be a tiny sliver, it is better to start with a thinner first row than get to the end and realize you are going to have a really thin last row.
Also be very careful when tapping the boards together, if you go to far you will chip/crush the board you are butting up to. When tapping the boards together be careful not to let the tapping block slip up, because that too will chip the plank.
Oh and don't forget to remove any baseboards before you start, and then measure again without the base boards.
When buying the underlay material make sure you get some that is made for concrete, you don't want moisture wicking up through the concrete into your new floor.
It's also a good idea to pick up an extra box or two of flooring, it could save you a trip to the store in the middle of the project if you underestimated or damage some of the planks during install. You can always return any unopened boxes when your done.
A miter saw is also very handy.
EDIT:
Be careful when tapping the planks into place, if you don't have weight on the floor or you are not using spacers the whole floor could shift and become crooked (I learned this the hard way). I was about halfway across the room when I realized the planks that I had previously installed were shifted, if I hadn't caught it I would have ended up with a very crooked floor.
You need a heavy duty construction adhesive. PL, Loctite Power Grab, Liquid Nails, etc.
PL makes a trim adhesive that should work well on that brick - it sets up kind of slow but once it gets set up it will stick that trim to brick just fine.
Best Answer
The boards near the wall should be replaced before you install trim, if possible. No repair will completely hide the damage.
Most manufacturers provide a means of replacing interior boards. Consider that for a professional result.
Otherwise, the boards in the field could be touched up with some enamel paint. You'll have to experiment with a base color and accent/grain colors. A matte varnish (urethane) might help match sheen, which is a big part of the challenge.