Good question , but would be nice if you provided some personal travel back ground .Have you ever visited Mexico on a bike ,done any travel in the country at all, speak any Spanish, how many days do you really have available ???
https://www.google.ca/maps/search/Ma...5,13.99z?hl=en
STUDY the above map and you can easily plot an interesting route through town to get you onto the highway Mex 101 which is the route you will want to follow south for a a few hours .
Look at this link to the google map of downtown Matamoros and Brownsville
There are several crossings to choose from .
If you like exploring the city centers then cross on the International Blv bridge in downtown Brownsville .
If you aim to get out of town as quickly as possible (which is a practice I do not recommend ) then just follow US 83 to its end and cross on that bridge .
In both cases you will find a building with the services to get the FMM (tourist card and stamp in your passport) and in the same buildings the Banjercito desk where you will need to get the TVIP and pay the REFUNDABLE TVIP security deposit.
Do not think you are done after pushing the button which gives you a green light to GO or a red light for a short inspection .
YOU MUST stop to get these documents on your own volition. The Mexican officials will not come running out to stop you and babysit you through this processing .
It is a very relaxed approach to border security. As has already been mentioned ,you may be the only persons you see getting these documents .
Your plan to go south for a few days and then head back is totally fine but do take the effort to retreat by way of a different set of roads .Don't go south, buy a tee shirt and then ride back the same way you came.
If you head south on the 101 it will eventually south of San Fernando bring
you to the junction where Mex 180 splits off to the south and to Tampico .
You will nowhere see any coast , the road is far inland until you hit Tampico
After the Rio Grande Valley farmlands it is mostly flattish , slightly rolling chaparral country which changes to more agricultural , some distant
mountains to see.
If you want a more mountain scenery then stick to 101 as far as Victoria
From there start the trek back north along the mountain front on Mex 85
Lots of side trips possible into the mountains if you have time
This route has plenty to see and slow you down to fill all available time