All translations from Alhatorah.org, unless noted otherwise.
As you mentioned, the getting up early in the morning and saddling a donkey:
Bilam:
וַיָּ֤קׇם בִּלְעָם֙ בַּבֹּ֔קֶר וַֽיַּחֲבֹ֖שׁ אֶת־אֲתֹנ֑וֹ וַיֵּ֖לֶךְ עִם־שָׂרֵ֥י מוֹאָֽב׃
Balaam rose up in the morning, and saddled his donkey, and went with the princes of Moab. (Numbers 22:2)
Avraham:
וַיַּשְׁכֵּ֨ם אַבְרָהָ֜ם בַּבֹּ֗קֶר וַֽיַּחֲבֹשׁ֙ אֶת־חֲמֹר֔וֹ וַיִּקַּ֞ח אֶת־שְׁנֵ֤י נְעָרָיו֙ אִתּ֔וֹ וְאֵ֖ת יִצְחָ֣ק בְּנ֑וֹ וַיְבַקַּע֙ עֲצֵ֣י עֹלָ֔ה וַיָּ֣קׇם וַיֵּ֔לֶךְ אֶל־הַמָּק֖וֹם אֲשֶׁר־אָֽמַר־ל֥וֹ הָאֱלֹהִֽים׃
Abraham rose early in the morning, and saddled his donkey, and took two of his young men with him, and Isaac his son. He split the wood for the burnt offering, and rose up, and went to the place of which God had told him. (Genesis 22:3)
Both of them took two lads with them:
Bilam:
וַיִּֽחַר־אַ֣ף אֱלֹהִים֮ כִּֽי־הוֹלֵ֣ךְ הוּא֒ וַיִּתְיַצֵּ֞ב מַלְאַ֧ךְ י״י֛ בַּדֶּ֖רֶךְ לְשָׂטָ֣ן ל֑וֹ וְהוּא֙ רֹכֵ֣ב עַל־אֲתֹנ֔וֹ וּשְׁנֵ֥י נְעָרָ֖יו עִמּֽוֹ׃
God’s anger was kindled because he went; and the angel of Hashem placed himself in the way for an adversary against him. Now he was riding on his donkey, and his two servants were with him. (Numbers 22:22)
Avraham:
וַיַּשְׁכֵּ֨ם אַבְרָהָ֜ם בַּבֹּ֗קֶר וַֽיַּחֲבֹשׁ֙ אֶת־חֲמֹר֔וֹ וַיִּקַּ֞ח אֶת־שְׁנֵ֤י נְעָרָיו֙ אִתּ֔וֹ וְאֵ֖ת יִצְחָ֣ק בְּנ֑וֹ וַיְבַקַּע֙ עֲצֵ֣י עֹלָ֔ה וַיָּ֣קׇם וַיֵּ֔לֶךְ אֶל־הַמָּק֖וֹם אֲשֶׁר־אָֽמַר־ל֥וֹ הָאֱלֹהִֽים׃
Abraham rose early in the morning, and saddled his donkey, and took two of his young men with him, and Isaac his son. He split the wood for the burnt offering, and rose up, and went to the place of which God had told him. (Genesis 22:3)
God sends angels to stop both of them:
Bilam:
וַיִּֽחַר־אַ֣ף אֱלֹהִים֮ כִּֽי־הוֹלֵ֣ךְ הוּא֒ וַיִּתְיַצֵּ֞ב מַלְאַ֧ךְ י״י֛ בַּדֶּ֖רֶךְ לְשָׂטָ֣ן ל֑וֹ וְהוּא֙ רֹכֵ֣ב עַל־אֲתֹנ֔וֹ וּשְׁנֵ֥י נְעָרָ֖יו עִמּֽוֹ׃
God’s anger was kindled because he went; and the angel of Hashem placed himself in the way for an adversary against him. Now he was riding on his donkey, and his two servants were with him. (Numbers 22:22)
Avraham:
וַיִּקְרָ֨א אֵלָ֜יו מַלְאַ֤ךְ י״י֙ מִן־הַשָּׁמַ֔יִם וַיֹּ֖אמֶר אַבְרָהָ֣ם ׀ אַבְרָהָ֑ם וַיֹּ֖אמֶר הִנֵּֽנִי׃
The angel of Hashem called to him out of the sky, and said, “Abraham, Abraham!” He said, “Here I am.” (Genesis 22:11)
There are more and more and more details shared between the two stories, which can be expanded upon detail by detail (and I'm sure they are by some):
- both of them are stopped before they can murder their companion (Isaac/the Donkey)
- there are mentions of Olah offerings that play significant roles in each story
- both ascend mountains
- The Jewish people are blessed by G-d at the conclusion of both stories
They are clearly understood to be a contrast to each other. The Talmud, in Sanhedrin 105b, tells us:
It is stated: “And Balaam rose in the morning and saddled his donkey” (Numbers 22:21). It was taught in a baraita in the name of Rabbi Shimon ben Elazar: Love negates the standard conduct of those of prominence. This is derived from Abraham, as it is written: “And Abraham rose early in the morning and saddled his donkey” (Genesis 22:3). Atypically, he saddled the donkey himself and he did not wait for his servants. Likewise, hatred negates the standard conduct of those of prominence. This is derived from Balaam, as it is stated: “And Balaam rose early in the morning and saddled his donkey” (Numbers 22:21).
Avot 5:19 presents a similar picture (see here).
In general, Avraham is given a message by God to perform a task that he will incur great loss from, and he goes selflessly to do so. Bilam is repeatedly told by God that he should not embark on this mission which he stands to benefit greatly from, and he goes nevertheless.