1/2 bath to 3/4 bath home improvement

I own a 54 year old house in Los angeles and I want to convert my 1/2 bath to a 3/4 bath by adding an upright shower. The room is "L" shaped with a toilet at one end of the "L" and a sink at the other. The idea is to make the room square by demo'ing the closet that makes the room "L" shaped. I have very little experience but a lot of guts. I have a professional who will hang the drywall after I am done installing the shower. My biggest concern is tapping into the existing plumbing. My plumbing questions are: What supplies will I need? How difficult is this project? How do I do it?
Thank you,
Hopelessly Cheap

Well just run down to Home Depot buy a shower stall, a valve, a bunch of pipe and fittings, come home and put it all together... It's easy!

The most difficult part is draining the shower and then venting the drain. You'll need access to an unfinished ceiling below. You'll have to do considerable exploratory work in order to plan this. Its a lot easier if you have at least some experience with previous installations. But its far from the rocket science that some make it out to be.

Really you have a bunch of planning to do before you go anywhere with this project. This involves specific information involving your building, The framing of it and the existing drains. Getting the drains and vents right will be the most difficult part. You can't expect to drop in and have us make a shopping list for you. You also have to look at what skills you have and know when something is within your abilities. Southern Man is a great example of a DIYer with little knowledge, limited skills and No Fear! Are you ready to do some work?

shower

Why not pay a professional to install the shower and hang the drywall yourself? IT would make more sense.

Hey there Southern Man,
I jist wanna lit ya-awl no I stopped at yur place an used yer toilit facility...
I done stopped er up pretty bad...

I guess the truth is out regarding Southern Man's qualifications on bathroom projects...

Your half-bath is plumbed for only two fixtures and therefore probably only has 1/2" supply lines.
These will have to be replumbed from the nearest 3/4" hot and cold supply lines with 3/4" to the bathroom to add a shower.
A shower will require a 2" trap and drain, vented after the trap to the main drain/waste/vent stack. If the half-bath doesn't have an exhaust vent, you also should add one. Showers put out a tremendous amount of moisture.
The expense and difficulty of this depends on your type of pipes (supply and drain), how much wall and/or ceiling demolition you need to do the plumbing, the remodeling that is necessary, and the shower unit that you choose. Adding a shower bath is a good investment, as bathrooms and kitchens have the highest return whenever the house is sold.
Before you do anything, contact your Building Inspection Department for local codes, permit and inspection requirements. Do NOT do this without a permit.
You may be required to use a licensed plumber and maybe a licensed electrican because you may well run into electrical lines in the walls or have to add an exhaust vent. Depends on local codes. You can do all demolition yourself. If allowed, you can do all of the other work, and we'll be glad to help you with it.
Good luck!