Generally, pray Sunnah Rakat / Naafil (almost all) at home is the best place. Based on hadith from Zaid ibn Tsabit radhiyallahu'anhu, Rasulullah shallallaahu`alaihi wa sallam said,

"The best prayer of a person is that performed in his home, with the exception of the obligatory prayer." [Bukhari 731, Muslim 1861]

So, pray Naafil is better to do in your house. Even in Al-Masjid al-Ḥaram in Makkah, if you're lived in there, then it's better to pray Naafil in your house. What's the evidence? Rasulullah shallallaahu`alaihi wa sallam and his sahabah radhiyallahu'anhum pray Naafil in their home, although they're know, pray in the mosque rewarded 100.000x.

But, there is some Naafil that needs to do in the mosque (and outside, on the field), for example Tarawih, Eid Fitr, Eid Adha, etc.

What is the benefit of praying Sunnah at home than mosque?

Simply, the more we follow sunnah, it always gives us more reward from Him.

Ibn Qudaamah said:

"Offering voluntary prayers at home is preferable, because praying at home is closer to sincerity and furthest removed from showing off. It is an act that is done in secret, whereas doing it is the mosque is in public, and secrecy is better."

Should I Pray only Fajr at mosque?

I don't know what do you mean.

What to do when i am traveling?

If you're travelling, you're not in your home. So, as far as i know, you'll have more reward if you pray at mosque. Even in Al-Masjid al-Ḥaram, it's better to pray Naafil in mosque. Because pray at Al-Masjid al-Ḥaram, rewarded 100.000x from ordinary mosque.

Generally, pray Sunnah Rakat / Naafil (almost all) at home is the best place. Based on hadith from Zaid ibn Tsabit radhiyallahu'anhu, Rasulullah shallallaahu`alaihi wa sallam said,

"The best prayer of a person is that performed in his home, with the exception of the obligatory prayer." [Bukhari 731, Muslim 1861]

So, pray Naafil is better to do in your house. Even in Al-Masjid al-Ḥaram in Makkah, if you're lived in there, then it's better to pray Naafil in your house. What's the evidence? Rasulullah shallallaahu`alaihi wa sallam and his sahabah radhiyallahu'anhum pray Naafil in their home, although they're know, pray in the mosque rewarded 100.000x.

But, there is some Naafil that needs to do in the mosque (and outside, on the field), for example Tarawih, Eid Fitr, Eid Adha, etc.

What is the benefit of praying Sunnah at home than mosque?

Simply, the more we follow sunnah, it always gives us more reward from Him.

Ibn Qudaamah rahimahullah said:

"Offering voluntary prayers at home is preferable, because praying at home is closer to sincerity and furthest removed from showing off. It is an act that is done in secret, whereas doing it is the mosque is in public, and secrecy is better."

Should I Pray only Fard at mosque?

Generally, yes. But, it always depends on the situation that we've faced. For example, some people doing Naafil in the mosque, because when they're coming back to home, they feel tired, so they're lazy to do Naafil. Or maybe after the Fard, they have a lot of work that needs to do. So, it's better to pray Naafil in the mosque, if you can't / lazy to do it in your home.

What to do when i am traveling?

If you're travelling, you're not in your home. So, as far as i know, you'll have more reward if you pray at mosque. Even in Al-Masjid al-Ḥaram, it's better to pray Naafil in mosque. Because pray at Al-Masjid al-Ḥaram, rewarded 100.000x from ordinary mosque.