Praise be to Allah.
Firstly:for guiding you to Islam; we ask Allah to make us and you steadfast in adhering to His religion.
It is better for a woman to pray in her house than in the mosque.
But, it may be better for her to pray in the mosque in some cases, such as if she has not memorised anything of the Holy Qur’an, or she cannot recite it well, or praying in the mosque is more motivating for her, or if she will benefit from classes in the mosque. In that case there is nothing wrong with her going out to the mosque to pray, so long as she is completely modest and covered.
Shaykh Ibn Baaz (may Allah have mercy on him) was asked: What is the ruling on a woman praying Taraweeh in the mosque?
The basic principle is that it is preferable and better for a woman to pray in her house, but if she thinks that there is some interest to be served by praying in the mosque, so long as she is covered and conducts herself properly, because it is more motivating for her, or because she will listen and benefit from the lessons, then there is nothing wrong with that, praise be to Allah, and it is also good because of the great benefits in that and because it encourages one to do righteous deeds. End quote.
He was also asked:
Is it permissible for a woman to pray Taraweeh in the mosque with the men?
Yes, that is mustahabb in her case if she is afraid that she will become lazy at home, otherwise her house is better. But if there is a need for that, then there is nothing wrong with it. The women used to offer the five daily prayers with the Prophet (blessings and peace of Allah be upon him), although he said: “Their houses are better for them.”
But some women may become lazy and lose motivation at home, so if a woman goes out to the mosque, properly covered, wearing hijab and avoiding making a wanton display, for the purpose of praying and listening to the people of knowledge, she will be rewarded for that, because this is a good purpose. End quote.
Fataawa Noor ‘ala ad-Darb (9/489)
Based on that, it is preferable in your case to pray Taraweeh in the mosque, but you should go out to the mosque fulfilling the conditions stipulated in sharee‘ah, of conducting yourself properly, covering properly, not wearing perfume, and so on.
Tahajjud is (voluntary) prayer at night. Some scholars stipulated that it refers to (voluntary) prayers at night after having slept.
Taraweeh refers to prayers at night in Ramadan.
See also the answer to question no. 143240
And Allah knows best.