Matamoros is situated on the United States-Mexico border across the Rio Grande River from the city of Brownsville, near where the Rio Grande empties into the Gulf of Mexico. It is one of several major land crossings between the United States and Mexico.
If crossing from the U.S. by foot, all three bridges between Brownsville and Matamoros charge a US$1 toll (and a US$0.25 toll to return). The turnstile is on the right side of the vehicle lanes in either direction.
You must bring a passport if you plan to return to the U.S., though you can enter the border zone of Mexico without it and will almost certainly not be asked to present documents. Mexican customs or the military or both may ask to search your bags, if you are carrying something. Note that in Brownsville U.S. customs officials do sometimes stop pedestrians heading for the bridge crossing and question them.
The wait to enter Mexico is negligible, but if you are planning to walk back across the bridge to the U.S. you might want to time it so that you do not have to stand for 30 minutes in extreme heat and humidity.
The bridge furthest east, the Veterans International Bridge at the end of HWY 77 on the Texas side, is not recommended for pedestrian crossing; it was designed for long-haul trucks and is a much longer bridge (and thus longer walk) than the other two. The bridge at International Blvd. is the shortest walk and more accessible to downtown Matamoros upon crossing.
If you follow International Blvd. south out of Brownsville, you'll cross over a short toll bridge (~US$2) and immediately be on Av. Alvaro Obregón. Border customs and security in the U.S.-to-Mexico direction is light, at worst a cursory glance-over, after which you'll immediately find yourself plunged into the streets and traffic of Matamoros.
If you're only planning to visit the "border zone," an area extending roughly 25 kilometers (15 miles) south from the border, you won't need a vehicle importation permit or a tourist card. If you intend to venture farther into Mexico, however, it's easiest to obtain your vehicle importation permit first thing at the border. Right after crossing the international bridge (literally at the foot of the bridge), you'll see a white building on your right housing all offices you may need to deal with: the Mexican Tourist office, Banjercito branch, and several Mexican automotive insurance (Seguros de Autos) vendors.
There are several car rental agencies. Rent something modest and unassuming.
Public transportation system is among the worst in Mexico. There are yellow microbuses called MaxiTaxi that charge about 1/2 dollar for a ride. The main station is located by the Brownsville/Matamoros Bridge. The "Sección 16" route will take you downtown.
Taxi cabs are among the most expensive in the country. Only the ones based on the bridge and the ones based around Plaza Hidalgo are legitimate. Expect to pay at least 10 dollars upon getting in.
There are hundreds of illegal taxi cabs that will charge you about a dollar to take you down their own pre-fixed routes or about 10 dollars wherever in town you want them to take you. These illegal cabs have no insurance and drive horribly. Do not trust them at all.
There is no Uber nor Cabify service.
Playa Bagdad is probably the least pretty beach in Mexico. There are no hotels there. Be warned: in Mexico beaches are considered Federal property, so the toll booth that the city government has installed has no right to charge you anything. They will say that entrance is about 25 pesos. If you ask if this is mandatory, they are forced to say no. You do not, repeat, not have to pay to get into the beach.