Mexico
 

Are there any tried and tested routes to take for a trip through Eastern Mexico? (from Texas to Belize)

Also, are there any good spots you guys reccomend for sight seeing along the way? (We would like to see the 'Cave of the Swallows'. )

E

If you want to see Sotano de las Golondrinas you pretty well nailed your route south from the border to being Mex 85 which runs right by Aquismon , start point for the trip to the cave. And a Beautiful route it is too- even if you don't go to the cave. South of Aquismon it is lovely mountains ,tropical tall forest changing into dry interior semidesert.
Mex 85 can then deliver you to Mex DF and your choice of routes farther to Belize is wide open, many alternatives await. Depends on your time available.
North of the DF , south of Pachuca you can take the side trip to Teotihuacan, impressive site of giant pyramids, good for a half day + wanderig around. From there you could avoid DF altogether by riding along its east flank on Mex 136 and Mex 115 south to Amecameca , view Volcan Popo', and onward to Izucar de Matamoros to pick up Hiway Mex 190 and mountain spine south to Oaxaca (Monte Alban ruins outside city, picturesque colonial city core) and then onward ( Mitla ruins) At Tehuantepec turn onto Mex 185 back to north coast for rest of run to Belize on Mex 180- 186 ( 40km side trip to Palenque )
OR after Teotihuacan head towards the DF as described above but from Mex 136 turn east on Mex 150/ 190 Libre for the nice ride over the pine mountain crest and go to Puebla ( interesting old city center, nearby Cholula ruins site ,west side) Stay on Mex 150 Libre all the way to Orizaba and Cordoba then out to the Gulf coast just south of Veracruz ( an interesting city center ) then east on mex 180 , varied green hill and coastal dune landscape, very scenic, to Minatitlan, Coatzacoalcos and Villahermosa and then on Mex 186 east past turnoff to Palenque and on to Belize as above..
Alternatives abound : From Puebla you can pick Mex 190 south which will give, on a clear day or early morning, a spectacular view of the east flanks of steaming,maybe erupting ,Volcan Popocatepetl (Popo') ,then you can carry on to Oaxaca.
From Mex 150 ,east of Cordoba you could take Mex145 southeast though drier sugarcane and pinapple gowing regions ,to Mex 185 and back up to Mex 180 to Minatitlan etc
In short ,if you have a basic map ANY shown connecting roads from your border crossing point to Belize will be worth a try.Of course you will have the return route also , so you can do a loop to cover as much as you can.You might consider looping from Palenque to Tenosique and the new border crossing at El Ceibo into Guatemala 's Peten ,thence to San Benito /Flores and Tikal and then into Belize.