We find in a Hadeeth that the Prophet, sallallaahu 'alayhi wa sallam, set out for Hajj on the 1st of Dhul-Hijjah while in another Hadeeth we find that he set on the 25th of Dhul-Qa'da. Can you please explain?
Narrated Aisha:
On the 1st of Dhul-Hijja we set out with the intention of performing Hajj. Allah's Apostle said, "Anyone who likes to assume the Ihram for 'Umra he can do so. Had I not brought the Hadi with me, I would have assumed the Ihram for 'Umra." Some of us assumed the Ihram for 'Umra while the others assumed the Ihram for Hajj. I was one of those who assumed the Ihram for 'Umra. I got menses and kept on menstruating until the day of 'Arafat and complained of that to the Prophet. He told me to postpone my 'Umra, undo and comb my hair, and to assure the Ihram of Hajj and I did so. On the right of Hasba, he sent my brother 'Abdur-Rahman bin Abi Bakr with me to At-Tah'im, where I assumed the Ihram for 'Umra in lieu of the previous one. Hisham said, "For that ('Umra) no Hadi, fasting or alms were required. (Al-Bukhaari)
Narrated Abdullah bin Abbas,
"The Prophet with his companions started from Medina after combing and oiling his hair and putting on two sheets of Ihram (upper body cover and waist cover). He did not forbid anyone to wear any kind of sheets except the ones colored with saffron because they may leave the scent on the skin. And so in the early morning, the Prophet mounted his mount while in Dhul-Hulaifa and set out till they reached Baida', where he and his companions recited Talbiya, and then they did the ceremony of Taqlid (which means to put the colored garlands around the necks of the Budn (camels for sacrifice). And all that happened on the 25th of Dhul-Qa'da. And when he reached Mecca on the 4th of Dhul-Hijja he performed the Tawaf round the Ka'ba and performed the Tawaf between Safa and Marwa." (Al-Bukhaari)

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

We have not come across any narration which states that the Prophet went out for Hajj on the first of Thul-Hijjah, so we do not know from where you got this narration. However, it is reported that 'Aa'ishah said about the farewell Hajj, “We went out around the beginning of Thul-Hijjah…” This does not mean that they went out on the first of Thul-Hijjah, rather the meaning is that they went out when the crescent was nearly due to appear. Ibn Hajar said, “This means when it was nearly time for the crescent to appear, and she previously stated 'we went out during the last five days of Thul-Qi'dah', and the five last days from the month are close to the end of the month, and so they were approaching the appearance of the new moon while they were on their way because they entered Makkah on the fourth of Thul-Hijjah.”

Allaah Knows best.