How does a Buddhist care for others' suffering?

"We’re going to have depend on one another. And one way to make sure that other people will be happy to help you is that you help them. You keep them in mind. Here’s why I keep saying: At the end of the day when you’ve done what you have to do, don’t just think about what you want to do now. Ask yourself: “Do I have a little energy to do at least one more thing for someone else’s well-being?” Make that a habit. That kind of habit then becomes something you can depend on." ~ Thanissaro Bhikkhu "Owners of Our Actions" https://www.dhammatalks.org/Archive/Writings/CrossIndexed/Published/Meditations10/200405_Owners_of_Our_Actions.pdf

I guess it broadly happens in three ways.

1. There's a focus on behaving ethically or virtuously -- keeping the precepts (or monks have stricter rules). These include no killing, no stealing, no lying, and more (e.g. no seducing their family).

   That protects other people from misbehaviour.

   More broadly there's some degree of emotional control as well as behavioural control. So "no anger" and "no coveting" as well as "no killing" and "no stealing".

   And IMO that helps protect other people from emotional contagion (of afflictive emotions).

2. There's helping people in mundane ways. Being a good member of the family and of society, friend and employer. Earning an honest living. Being prudent, being generous, being thrifty not dissipated.

   In that way you're able to help people in some worldly ways.

3. You're better able to help other people find their own liberation, perhaps by formal teaching or more generally perhaps by simply behaving (practising) as a good example.

There may be a step beyond that i.e. "bodhisattva" -- which is not a feature of the form of Buddhism which I've studied so far, maybe the three ways listed above are at least a good beginning, possibly sufficient.

Does the Buddhist value more attaining enlightenment for himself or helping others in need?

That question might assume things about identity which may be non-obvious from a Buddhist perspective -- Buddhism identifies selfishness as a problem, a cause or even the cause of "suffering".

Someone might answer that enlightenment is helping all beings (i.e. that the question is a false dichotomy).

Some of the doctrine in the suttas is like,

Even as a mother protects with her life
Her child, her only child,
So with a boundless heart
Should one cherish all living beings;
Radiating kindness over the entire world:

There's also What is the difference between 'compassion' and 'pity'?

Buddhism also mentions intention (what you're trying to do) and skill (how well or successfully you do it).

I think some of the elementary doctrine includes "skillful virtue results in lack of remorse" -- and "helping others in need" might belong to the category of actions which you don't regret -- and therefore helping someone is a little taste of enlightenment.

Together with "selfishness", "greed" is a problem (there are other problems too). I think that generosity is recommended even as part of the elementary training in virtue.

Also what if others do t have the skills to practice to go towards enlightenment?

I'd like to hope that everyone has the skill.

I'm not sure everyone has the right environment or circumstance (or society), though, maybe history (training) too -- perhaps their environment is part of what you affect (improve) by your own practice.

When a person is sick, due to knowing they are sick, they act to visit a doctor. To be a doctor, a person must study medicine at university for at least 5 years, before they can help sick people.

Similarly, a person who is suffering must, via their own initiative, seek out a solution. Similarly, to be a Buddhist that can help others end suffering, a person must train for enlightenment.

An enlightened Buddhist makes themselves available to help others. However, others must learn to help themselves by following the advice of Buddhism.