Public courts are rare in Mexico. Mexico has a lot of private clubs, where guests are usually welcome. These clubs don't do much in the way of publicizing their existence, so you'll have to do some asking around.
“Deportivos” (sports clubs) are more accessible. Usually, you pay a day fee of 100 to 150 pesos at the entrance. This fee covers all the club's facilities, so bring your bathing suit. The “Club Britania”'s found in many Mexican cities are actually deportivos.
Ball boys are commonplace wherever tennis is played. This is something easy to get used to; they speed up the game considerably. You're expected to tip them; the going rate is ten or 15 pesos per player per set.
If you're going to a high-altitude place such as Mexico City, Oaxaca de Juarez, or San Miguel de Allende, don't bother to bring balls with you. Only pressureless (“sin presión”) balls are legal in these areas. The most common brand is Tre-nis (sold as Tretorn in the US and Canada), but pressureless balls are easy to spot on store shelves; they're sold in cardboard boxes instead of cans.
There are clay courts (“canchas de arcilla”) in Mexico, but they're gradually going the way of the dinosaur, due to maintenance costs. Hard courts are called “canchas duras”. You may come across synthetic turf courts; I suggest trying to avoid them.