Kit

We start with the same kit as we would at an indoor climbing wall:

Harness
For sport climbing you need a couple of gear loops (it's best to rack quickdraws a few on each side to make it easier to clip.
Shoes
Again, if you're just getting started outside you could use your indoor shoes but remember you'll be wearing them a lot longer than you would inside so a more comfortable paris is a good idea. Laces provide a good fit but are a pain to slacken off e.g. while belaying.
Chalkbag
Belay device and carabiner
Use whatever you are comfortable with, an assisted breaking device (such as a Petzl GriGri or Mammut Smart) is highly recommended. There can be a lot of hanging on the rope when you first climb outside!
Rope 60m
While you could use your wall rope you will most probably need a longer rope. A 60m will let you climb routes up to about 27m high.

The additional equipment:

Helmet
This is entirely your choice but I implore you to wear one. Loose rock and falling equipment is a very real hazard. An incapacitated belayer not going to hold the rope.
Quickdraws × 10
Some guide books will tell you the number of bolts, however you will need an additional 2 quick draws for the anchor and it’s always good to have a couple spare in case you drop one or have a miss-hap of some kind.
Screw gate carabiners × 2
One for the Cow’s tail and one for the figure of 8 on a bight.
Cow’s tail
To anchor ourselves to the top of a climb while cleaning the anchors. This can be as simple as a 60cm sling knotted to the right length or a specialised piece of equipment like the Metolius PAS (Personal Anchor System) or Petzl Connect Adjust Lanyard. I would recommend against using a stitched daisy chain, these are designed for Aid climbing and are only rated end to end, not for clipping into individual pockets.
Towel or old cloth
Give your shoes a wipe down before the climb
Rope Bag
The base of the climb can be a dusty, muddy, or sandy place. Keep that rope as clean as you can.

Knots

Lark’s foot / Girth hitch
Attach cow’s tail to harness
Figure of 8 on a bight
Tie into the rope without needing the free end of the rope.

Skills & techniques

Clipping a bolt hanger
Attaching the quickdraw to the bolt hanger.
Cleaning the anchor
How to safely lower off a sport climb without leaving behind any equipment.