Avodah - What is it?

(ב) שִׁמְעוֹן הַצַּדִּיק הָיָה מִשְּׁיָרֵי כְנֶסֶת הַגְּדוֹלָה. הוּא הָיָה אוֹמֵר, עַל שְׁלשָׁה דְבָרִים הָעוֹלָם עוֹמֵד, עַל הַתּוֹרָה וְעַל הָעֲבוֹדָה וְעַל גְּמִילוּת חֲסָדִים:

(2) Shimon the Righteous was from the remnants of the Great Assembly. He would say, "On three things the world stands: on the Torah, on the avodah and on acts of lovingkindness."

What is the avodah in this quote?

(ד) על העבודה כיצד כל זמן שעבודת בית המקדש קיימת העולם מתברך על יושביו וגשמים יורדין בזמנן ... שאין עבודה שהיא חביבה לפני הקב״ה יותר מעבודת בית המקדש

On avodah how? As long as the avodah [service] of the Temple is maintained, the world will be blessed and the rains come down in their seasons. That there is nothing more loved by God than the avodah [service] of the Temple

What is avodah here?

(ב) תִּכּ֤וֹן תְּפִלָּתִ֣י קְטֹ֣רֶת לְפָנֶ֑יךָ מַֽשְׂאַ֥ת כַּ֝פַּ֗י מִנְחַת־עָֽרֶב׃
(2) Take my prayer as an offering of incense, my upraised hands as an evening sacrifice.

What does the Tanakh say about prayer and sacrifice here?

א"ר חייא בר אבא א"ר יוחנן כל הנפנה ונוטל ידיו ומניח תפילין וקורא ק"ש ומתפלל מעלה עליו הכתוב כאלו בנה מזבח והקריב עליו קרבן

On a similar note, Rabbi Ḥiyya bar Abba said that Rabbi Yoḥanan said: Anyone who relieves himself, washes his hands, dons phylacteries, recites Shema, and prays, the verse ascribes credit to him as if he built an altar and offered a sacrifice upon it.

What is the connection between prayer and sacrifice?

איתמר רבי יוסי ברבי חנינא אמר תפלות אבות תקנום רבי יהושע בן לוי אמר תפלות כנגד תמידין תקנום
The dispute between the Rabbis and Rabbi Yehuda with regard to the times beyond which the different prayers may not be recited is rooted in a profound disagreement, also manifest in a later amoraic dispute. It was stated: Rabbi Yosei, son of Rabbi Ḥanina, said: The practice of praying three times daily is ancient, albeit not in its present form; prayers were instituted by the Patriarchs. However, Rabbi Yehoshua ben Levi said that the prayers were instituted based on the daily offerings sacrificed in the Holy Temple, and the prayers parallel the offerings, in terms of both time and characteristics.

How does this expand upon what the first Berachot text says about prayer and sacrifice?

איזו היא עבודה שהיא בלב הוי אומר זו תפלה וכתיב בתריה (דברים יא, יד)

...Which is the service of God that is performed in the heart? You must say that this is referring to prayer. And, afterward, it is written: “And I shall give the rain of your land in its due time, the first rain and the last rain” (Deuteronomy 11:14).

How does this text connect between avodah, sacrifice, and prayer?