新たな始まり is a Japanese word that means “new beginning” or “new start.”

Here's the character breakdown:
新た (arata) = new; fresh; novel; newly; freshly; or this can be like the prefix “re-” like “re-start” or “reset.”
な (na) is kind of a connecting article. This glues “new” to “beginning.”
始まり (hajimari) = origin; beginning.

Note: Because this selection contains some special Japanese Hiragana characters, it should be written by a Japanese calligrapher.

新的開始 literally means “new beginning” in Chinese characters.

The character means “new.”

The second is a possessive article connecting the ideas of new & beginning.

The last two characters can mean “to begin,” “beginning,” “to start,” “initial,” “commencement,” or “initiation.”

新しい始め is a verbose Japanese phrase that means “new beginning.”

The first three characters mean new, novel, fresh, recent, latest, up-to-date, or modern.

The last two characters mean beginning, start outset, opening, or origin.

Note: Because this selection contains some special Japanese Hiragana characters, it should be written by a Japanese calligrapher.

事始め is a Japanese word that means taking up a new line of work or the beginning of new things.

In another context, it can refer to starting the preparations for New Year's festivities or resuming work after the New Year's vacation.

伊始 is a short version of “new beginning” or simply “beginning” in Chinese characters.

You can also translate this as “from this moment on,” “starting now,” or “henceforth.”

In the day-to-day speech, this word can apply to starting a new job, beginning a new career, entering a new chapter of your life, or taking a new position (in politics, scholarship, etc.).

重新做人 is a Chinese proverb that means or suggests starting a new life or to make a fresh start.

啟盛世開太平 means “To bring flourishing peace and security to the world (our current era).”

It's a wish that a new door leading to peace and prosperity could be opened to mankind.

Character and word breakdown:
啟 to open; to start; to initiate; to enlighten or awaken.
盛世 a flourishing period; period of prosperity; a golden age.
開 to open; to start; to turn on.
太平 peace and security; peace and tranquility; peace; tranquility.
I don't like to do breakdowns like this, as the words altogether create their unique meaning (encompassed in the main title above).

同性戀自豪感 or “Gay Pride” is a new idea in China. It's so new, that we may have just started the movement by translating this phrase.

If they ever do start having gay pride parades in China, my best guess is that these 6 characters will constitute the term they use to title the parade/movement.

Who knows, maybe in 10 years they will have a pride parade march straight past Tian'anmen Square on Chang An Street (the main drag in Beijing).