All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, we say that there is no harm in traveling with them since your maternal and paternal uncles will be with you; as they are your Mahrams. Ibn ‘Abbaas narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said, "No woman should travel unless she is escorted by a male Mahram." [Al-Bukhaari and Muslim]
Your male cousins are not among your Mahrams, so you must abide by the Shariah provisions in talking to them. It is permissible if there is a need for it and within the scope of that need, and without using soft speech or the like that may lead to Fitnah (temptation). Allah, The Exalted, says (what means): {O wives of the Prophet, you are not like anyone among women. If you fear Allah, then do not be soft in speech (to men), lest he in whose heart is disease should covet, but speak with appropriate speech.} [Quran 33:32]
The permissibility that we mentioned applies in case Fitnah is not feared. However, if Fitnah is feared, then talking to them becomes prohibited. An-Nawawi quoted the following statement of Abu Sa‘eed Al-Mutawalli:
"If the woman is a non-Mahram and she is beautiful and there is fear that a man may be tempted by her, then he should not greet her. If he does, then it is not permissible for her to reply. She should not initiate the greeting either, and if she does, she does not deserve a response. If he responds, then this is disliked for him. If she is an old woman who would not tempt men, then it is permissible for her to greet the man and for the man to return her greeting. If there is a group of women, then a man may greet them, or if there is a group of men, they may greet a woman so long as there is no fear that any of the parties may be tempted."
We also warn of family gatherings in which Shariah provisions are not observed. The Muslim women should wear their hijaab before non-Mahram relatives, and there should be no prohibited mixing and the like. If someone is unaware of such provisions, he should read the scholars' statements in this regard. If someone knows of them and ignores them, he should kindly be advised. Firmness in this regard is required; it is impermissible for the Muslim to compromise his religion in order to please anyone or seek the pleasure of people by doing what incurs the Wrath of Allah, The Almighty. It was reported that ‘Aa'ishah wrote a letter to Mu‘aawiyah reading, "Peace be upon you. I heard the Messenger of Allah say, "Whoever seeks the pleasure of Allah by incurring people's wrath, Allah will suffice him from the people. And whoever seeks the people's pleasure by incurring the Wrath of Allah, He will entrust him to the people." [At-Tirmithi]
We also warn against the kinds of sports that may be inappropriate for women, such as climbing, which is mentioned in the question. For more benefit, please refer to fatwa 126934.
Allah knows best.