Escape the suffering?

The answer to What teachings do all schools of Buddhism share? includes,

5. We accept the Four Noble Truths taught by the Buddha, namely, Dukkha, the fact that our existence in this world is in predicament, is impermanent, imperfect, unsatisfactory, full of conflict; Samudaya, the fact that this state of affairs is due to our egoistic selfishness based on the false idea of self; Nirodha, the fact that there is definitely the possibility of deliverance, liberation, freedom from this predicament by the total eradication of the egoistic selfishness; and Magga, the fact that this liberation can be achieved through the Middle Path which is eight- fold, leading to the perfection of ethical conduct (sila), mental discipline (samadhi) and wisdom (pañña).

One answer to your question is that the 'way' to escape suffering is the "Fourth Noble Truth", which is the "Noble Eightfold Path", which is also known as the Threefold Training.

Another answer might the "taking refuge" in the Triple Gem:

2. We all take refuge in the Triple Jewel: the Buddha, our Teacher; the Dhamma, his teaching; and the Sangha, the Community of holy ones. In other words, we take refuge in the Teacher, the Teaching and the Taught.

Maybe the above are slightly Theravada-centric answers. The answer I accepted to this question, How to explain what Buddhism is?, reveals a range of other/different Buddhist schools/understandings also.

Someone like the Dalai Lama seems to preach compassion and a kind of selflessness, maybe thinking of all people as the same, as a way to escape suffering.

And something like the answer to this question, What is meant by Namu Amida Butsu and also Nam-myoho-renge-kyo? begins to explain another school of Buddhism. If you're familiar with the idea that, as you wrote in the OP, "escape is only through faith in Jesus Christ", this answer too depends on a kind of "faith" or realization.

The above is a theoretical or a referenced answer.

An answer more from personal experience (this site wants answers based on references and/or personal experience) is that there is some escape from suffering (when suffering arises) from having learned (i.e. read or hear, study, try to understand, make sense of, internalize, and apply) dhamma.

Consequently if a conceit like "I am experiencing suffering" arises, then some dhamma (or awareness of dhamma arises and) might provide some kind of mental, habitual antidote.

Examples of what I'm talking about are summarized as/by relatively short phrases. The phrases are short enough that they can be "held in mind" i.e. I can remember a phrase as a unit. I haven't tried to remember a whole paragraph at once, but I find that a short phrase is helpful as mental furniture.

I won't try to explain every phrase in this answer (that would be wordy) but the following are some of those phrases or important summaries that I found helpful:

• Simsapa Sutta: The Simsapa Leaves (SN 56.31)

  Therefore your duty is the contemplation,

  'This is stress... This is the origination of stress... This is the cessation of stress.'

  Your duty is the contemplation,

  'This is the path of practice leading to the cessation of stress.'

• Anatta-lakkhana Sutta: On the No-self Characteristic (SN 22.59):

  Therefore, surely, O monks, whatever (of the five khandhas) ... all that ... must be regarded with proper wisdom, according to reality, thus:

  'This is not mine, this I am not, this is not my self.'

• Kimattha Sutta: What is the Purpose? (AN 11.1)

  Skillful virtues have freedom from remorse as their purpose, Ananda, and freedom from remorse as their reward.

• Brahmana Sutta: To Unnabha the Brahman (SN 51.15):

  Whatever ... he first had ..., on attaining ... that ... is allayed. This is a path with an end

People also recommend good company (friends, teachers) and good communities as helpful; for example:

• Upaddha Sutta: Half (of the Holy Life) (SN 45.2)

  Don't say that, Ananda. Don't say that. Admirable friendship, admirable companionship, admirable camaraderie is actually the whole of the holy life. When a monk has admirable people as friends, companions, & comrades, he can be expected to develop & pursue the noble eightfold path.

  But also:

  It is in dependence on me as an admirable friend that beings subject to birth have gained release from birth

See also, for example, this comment.

It's not easy in an answer to answer from the whole of Buddhism; so the above is biased on my personal experience. See also for example this topic (Tibetan) and/or maybe this topic (Zen).