What punishment do people get for eating non-vegetarian food ? Do they get punished in hell or do they get punished in this world ?
Bhishma told this to Yudhishthira that a person who eats non-vegetarian food does go to hell.
One should never eat meat of animals not dedicated in sacrifices and that are, therefore, slain for nothing, and that has not been offered to the gods and Pitris with the aid of the ordinances. There is not the slightest doubt that a person by eating such meat goes to Hell.
That wretch among men who, following the path of religious rites and sacrifices laid down in the Vedas, would kill a living creature from desire of eating its flesh, would certainly become a resident of hell.
Meat that is sanctified is pure. Others are not.
Manu himself has said that meat which is sanctified with mantras and properly dedicated, according to the ordinances of the Vedas, in rites performed in honour of the Pitris, is pure.
All other meat falls under the class of what is obtained by useless slaughter, and is, therefore, uneatable, and leads to Hell and infamy. One should never eat, O chief of Bharata's race, like a Rakshasa, any meat that has been obtained by means not sanctioned by the ordinance. Indeed, one should never eat flesh obtained from useless slaughter and that has not been sanctified by the ordinance.
King Vasu had to suffer for his mistake.
It is heard that in the ancient Kalpa, persons, desirous of attaining to regions of merit hereafter, performed sacrifices with seeds, regarding such animals as dedicated by them. Filled with doubts respecting the propriety of eating flesh, the Rishis asked Vasu the ruler of the Chedis for solving them. King Vasu, knowing that flesh is inedible, answered that is was edible, O monarch. From that moment Vasu fell down from the firmament on the earth. After this he once more repeated his opinion, with the result that he had to sink below the earth for it.
What happens if you abstain from it?
The seven celestial Rishis, the Valakhilyas, and those Rishis who drink the rays of the sun, endued with great wisdom, applaud abstention from meat. The Self-born Manu has said that that man who does not eat meat, or who does not slay living creatures, or who does not cause them to be slain, is a friend of all creatures. Such a man is incapable of being oppressed by any creature. He enjoys the confidence of all living beings. He always enjoys, besides, the approbation and commendation of the righteous.
Vrihaspati has said that that man who abstains from honey and meat acquires the merit of gifts and sacrifices and penances. In my estimation, these two persons are equal, viz., he who adores the deities every month in a horse-sacrifice for a space of hundred years and he who abstains from honey and meat. In consequence of abstention from meat one comes to be regarded as one who always adores the deities in sacrifices, or as one who always makes gifts to others, or as one who always undergoes the severest austerities. That man who having eaten meat gives it up afterwards, acquires merit by such an act that is so great that a study of all the Vedas or a performance, O Bharata, of all the sacrifices, cannot bestow its like.
Hinduism recommends abstaining from eating meat as the act (of abstaining) bears spiritual results.
See the following verse from Manu which says that eating meat is human nature so as such there is nothing wrong in it but abstaining from it bears great fruits.
5.56. There is no sin in eating meat, in (drinking) spirituous liquor, and in carnal intercourse, for that is the natural way of created beings, but abstention brings great rewards.
But as regards punishments are concerned, then Manu also gives the punishments for those who consume meat unlawfully.
5.38. As many hairs as the slain beast has, so often indeed will he who killed it without a (lawful) reason suffer a violent death in future births.
The following verses are also relevant:
5.48. Meat can never be obtained without injury to living creatures, and injury to sentient beings is detrimental to (the attainment of) heavenly bliss; let him therefore shun (the use of) meat.
5.49. Having well considered the (disgusting) origin of flesh and the (cruelty of) fettering and slaying corporeal beings, let him entirely abstain from eating flesh.
5.51. He who permits (the slaughter of an animal), he who cuts it up, he who kills it, he who buys or sells (meat), he who cooks it, he who serves it up, and he who eats it, (must all be considered as) the slayers (of the animal).
5.52. There is no greater sinner than that (man) who, though not worshipping the gods or the manes, seeks to increase (the bulk of) his own flesh by the flesh of other (beings)
5.53. He who during a hundred years annually offers a horse-sacrifice, and he who entirely abstains from meat, obtain the same reward for their meritorious (conduct).
5.55. ’Me he (mam sah)’ will devour in the next (world), whose flesh I eat in this (life); the wise declare this (to be) the real meaning of the word ’flesh’ (mamsah)
The overall essence of the above verses is given by Manu in 5.56 (already given further above) and which is that meat eating is not entirely forbidden and there is nothing much "wrong" in it but abstaining from it is the recommendation of scriptures.
(Here is a closely related answer: https://hinduism.stackexchange.com/a/20584/4732)