I'm thinking of flying from New Zealand to LA instead of South America because I think it might be cheaper. This way I can travel down through Central America starting with Mexico.
Is there a bus or train I can catch from LA to Mexico and how much is it?
You can take a bus or train from LA to San Diego, but you have to go downtown to get both (which is a ways from the airport). It's probably cheaper to get a flight directly from LA to San Diego RT if you book at least 14 days ahead. (Book on Southwest Airlines).
Once you get to San Diego, you can get the Trolley line down to the border, then go across the border there. However, you will still have the problem of trying to get around Mexico. Mexico is a very large country, and unless you speak Spanish, you will need to stick to the main resorts or the big cities. Also, there's been a bit of crime (drug wars) going on in Sinaloa, which is the main road/bus route down into the heart of Mexico.
If you actually want to see more of Mexico, either fly to the city of your choice from Los Angeles, or get a rental car from San Diego. (If you take this last option, be sure to ask to buy "Mexican insurance" and make sure they will let you take their car across the border--some agencies will, some won't).
Just visiting the tourist area of Tijuana (at the border) will make you think there's nothing worth seeing in Mexico. Driving down to Rosarito and Ensenada, or flying down to Mexico City, Cancun, Acapulco, Puerto Vallarta, or Cabo San Lucas will have you wanting to go back again and again and again (as I've done).
The U.S. State Department has a travel alert for Baja California. The area between San Diego and Ensenada is particularly dangerous right now (drug war stuff). There have been several incidents of gringos getting caught in the crossfire. It's VERY dangerous - I wouldn't do it myself.
You might want to fly into Phoenix or Tucson in Arizona and drive into Mexico through Nogales (Sonora). That way you avoid Baja California completely. As Calcruzer noted, it is imperative that you get Mexican insurance and verify with the rental car agency that you can take the vehicle into Mexico. If you don't have Mexican insurance and you are involved in an accident, you will be taken to jail. Be aware that some car rental companies will let you drive into, but no further than 20 miles or so from the border. So definitely check all the details with them.
You could use the bus from Phoenix or Tucson and rely on buses and taxis within Mexico. I know it's easy and relatively cheap to take a bus from Phoenix into Mexico. TBC's bus service from Phoenix to Guaymas is about $40 one way. You might want to check with Greyhound. If you take the bus, remember to tell the driver you need to get a Mexican tourist card so they will stop at the 21Km checkpoint south of Nogales.
If you do decide to go through Sonora (the Mexican state that borders Arizona), be sure to stop at Magdalena and visit the Colossio memorial. Hermosillo is the state capital and it's worth a day tour. Further south in the state, Guaymas is interesting and San Carlos is nice but chock full of American retirees. Navajoa, which I liked, is the furthest south I have been in Mexico.
You will love Mexico. The people are wonderful!