There are many mythical beings mentioned in the Hindu scriptures some of whom may have some association with the creatures mentioned in the western mythologies:
SECTION LXVI of Sambhava Parva of Mahabharat tells about the progeny of Brahma's mind born sons and mentions the following beings:
"Vaisampayana said, 'It is known that the spiritual sons of Brahman were the six great Rishis. There was another of the name of Sthanu. And the sons of Sthanu, gifted with great energy, were, it is known, eleven. They were Mrigavayadha, Sarpa, Niriti of great fame: Ajaikapat, Ahivradhna, and Pinaki, the oppressor of foes; Dahana and Iswara, and Kapali of great splendour; and Sthanu, and the illustrious Bharga. These are called the eleven Rudras. It hath been already said, that Marichi, Angiras. Atri, Pulastya, Pulaha, and Kratu--these six great Rishis of great energy--are the sons of Brahman. It is well-known in the world that Angiras's sons are three,--Vrihaspati, Utathya, and Samvarta, all of rigid vows. And, O king, it is said that the sons of Atri are numerous. And, being great Rishis, they are all conversant with the Vedas, crowned with ascetic success, and of souls in perfect peace. And, O tiger among kings, the sons of Pulastya of great wisdom are Rakshasas, Monkeys,Kinnaras half-men and half-horses, and Yakshas. And, O king, the son of Pulaha were, it is said, the Salabhas the winged insects, the lions, the Kimpurushas (half-lions and half-men), the tigers, bears, and wolves. And the sons of Kratu, sacred as sacrifices, are the companions of Surya, the Valikhilyas, known in three worlds and devoted to truth and vows.
These half-men half-horses beings are very similar to the Centaurs in Greek Mythology and the half-lion half-man beings are actually reminiscent of some Egyptian goddesses such as Sekhmet.
SECTION CLXIX of the Shanti Parva also describes few more species:
At last coming upon a road that led towards the ocean he journeyed on till he reached a delightful and heavenly forest abounding in flowering trees. It was adorned with mango trees that put forth flowers and fruits throughout the year. It resembled the very woods of Nandana (in heaven) and was inhabited by Yakshas and Kinnaras. It was also adorned with Salas and palmyras and Tamalas, with clusters of black aloes, and many large sandal trees. Upon the delightful tablelands that he saw there, fragrant with perfumes of diverse kinds, birds of the foremost species were always heard to pour forth their melodies. Other winged denizens of the air, called Bharundas, and having faces resembling those of human beings, and those called Bhulingas, and others belonging to mountainous regions and to the sea, warbled sweetly there, Gautama proceeded through that forest, listening, as he went, to those delightful and charming strains of nature's choristers.
Yakshas, the forest spirits may actually be similar to Satyrs who are half human half beasts while Bharundas are just a tad bit like the Sphinx in the sense that they have human face and wings but no lion-body. Strictly speaking the Sphinx is known as the Purush-Mriga in Sanskrit and can be commonly seen in temples especially in the southern part of the country. This comes from a folktale:
Krishna called Bheem and asked him to invite/bring Purusha Mriga (a yogi dwelling in the Himalayas) for the grand yagna of Yuddhisthir. Purusha Mriga or Vyaghrapada was a very powerful yogi and Krishna warned Bheem - if he couldn't match Mriga's speed of mind, he would be killed.
Scared, Bheem proceeded towards Himalayas. En route, he met Hanuman who stopped him to ask the cause of worry in his face. Bheem told him the entire story to which Hanuman, his elder brother, told him that Purusha Mriga was a staunch Shiva devotee. The only way to stop him would be to place Shiva's lingam throughout the route so that he stops to worship the idol. That being said, he gave Bheem a bunch of feathers from the tip of his tail - dropping each feather would spawn a lingam - distracting Purush Mriga and helping Bheem.
Then of course there are the Nagas half human half snake beings and the Garudas with human-bird hybrid bodies. Both of them were born from eggs laid by Kashyap's human wives as described in the SECTION XVI of Astik Parva of Mahabharat
"Sauti continued, 'O best of regenerate ones, after a long time, Kadru brought forth a thousand eggs, and Vinata two. Their maid-servants deposited the eggs separately in warm vessels. Five hundred years passed away, and the thousand eggs produced by Kadru burst and out came the progeny.
Garud's younger brother Arun is also an interesting creature with a half-formed body:
But the twins of Vinata did not appear. Vinata was jealous, and therefore she broke one of the eggs and found in it an embryo with the upper part developed but the lower one undeveloped.....Thus cursing his mother, the child rose to the sky. O Brahmana, even he is the charioteer of Surya, always seen in the hour of morning!
"Then at the expiration of the five hundred years, bursting open the other egg, out came Garuda, the serpent-eater. O tiger of Bhrigu's race, immediately on seeing the light, that son of Vinata left his mother. And the lord of birds, feeling hungry, took wing in quest of the food assigned to him by the Great Ordainer of all.".
The Mahabharat Astik Parva SECTION XXXVI describes Shesh-Naga the chief of Nagas as a snake with a human-like body:
And the Grandsire of all, Brahma, saw that ascetic with knotted hair, clad in rags, and his flesh, skin, and sinews dried up owing to the hard penances he was practising. And the Grandsire addressing him, that penance-practising one of great fortitude, said, 'What is that thorn doest, O Sesha? Let the welfare of the creatures of the worlds also engage thy thoughts. O sinless one, thou art afflicting all creatures by thy hard penances. O Sesha, tell me the desire implanted in thy breast.'
"Unto Sesha who had said so, the Grandsire said, 'O Sesha, I know the behaviour of all thy brothers and their great danger owing to their offence against their mother. But O Snake, a remedy (for this) hath been provided by me even beforehand. It behoveth thee not to grieve for thy brothers. O Sesha, ask of me the boon thou desirest. I have been highly gratified with thee and I will grant thee today a boon. O best of snakes,? it is fortunate that thy heart hath been set on virtue. Let thy heart be more and more firmly set on virtue.'
The Harivansha also gives a description of Sheshnaga in a half human-half serpent form where he is shown to have attributes of Lord Balaram:
tasya madhye sahasrAsyaM hematAlochChritadhvajam |
lA~NgalAsaktahastAgraM musalopAshritodaram ||2-26-49
In the middle (of nAgaloka), with thousand heads, with a flag showing a golden palm leaf, carrying a plough in one hand (he saw sheSha). A club (musala) was kept near his belly.
asitAmbarasaMvItaM pANDuraM pANDurAsanam |
kuNDalaikadharaM mattaM suptamamburuhekShaNam ||2-26-50
Wearing a blue dress, with white complexion and seated on a white seat, wearing one ear ring, intoxicated, with eyes looking like sleepy lotus flowers,
bhogotkarAsane shubhre svena dehena kalpite |
svAsInaM svastikAbhyAM cha varAhAbhyAM mahIdharam ||2-26-51
seated on the white seat, created by his own serpentine body, adorned by svastika and varAha and holding the earth,
ki~nchitsavyApavR^ittena maulinA hemachUlinA |
jAtarUpamayaiH padmairmAlayAchChannavakShasam ||2-26-52
with his head slightly inclined to the left adorned with golden ornaments and his chest covered with a garland of golden lotus flowers,
raktachandanadigdhA~NgaM dIrghabAhumarindamam |
padmanAbhasitAbhrAbhaM bhAbhirjvalitatejasam ||2-26-53
the killer of enemies with his long arms smeared with paste of red sandal, the one with a lotus flower from his navel, with white complexion, attractive with his effulgence,
The Naga Takshak is also mentioned transforming from a human into a serpent:
"On the road Utanka perceived coming towards him a naked idle beggar sometimes coming in view and sometimes disappearing. And Utanka put the ear-rings on the ground and went for water. In the meantime the beggar came quickly to the spot and taking up the ear-rings ran away. And Utanka pursued the thief with the utmost speed. And having with great difficulty overtaken him, he seized him by force. But at that instant the person seized, quitting the form of a beggar and assuming his real form, viz., that of Takshaka, speedily entered a large hole open in the ground. And having got in, Takshaka proceeded to his own abode, the region of the serpents.
There are also other mythical beings such as certain rakshas species with humanoid bodies and animal faces. FOr example, Mahishasura the buffalo-demon is somewhat similar to the dreaded Minotaur of Greek myths:
Similarly, the Greek Hydra is a huge serpent that was defeated by Heracles and it can be equated with the Kaliya Naga of multiple hoods defeated by Krishna. BTW many of the labors of Hercules are also similar to the acts performed by Krishna in his childhood. Most of these stories may have come from common origins and hence we do find some similarities between the kind of mythical beings in both though each culture would have of course transformed the myth into its own version later on.