It strikes me as slightly difficult to understand that these were so difficult for Avraham. If Hashem revealed Himself to me, I would like to think that I would do whatever He asked me to do without question. All the more so Avraham, who was on an infinitely higher madreiga than I am on.

On the other hand, perhaps this is not absolutely true. Other people experienced Hashem and 'failed' - Adam Harishon, Noach (when he debased himself), even Moshe with the rock.

What were the intrinsic tests that Avraham had to face? The acts themselves, or the faith in Hashem?

Good questions, and ones to which a possible answer is given in the Torah Tidbits Sedra Summary at Torah Tidbits - Lech Lecha

It says as follows:

Ramban states that "Avra(ha)m sinned a great sin (inadvertently)" by leaving the land to which G-d had sent him (even though there was a famine) and by risking Sarai's life and honor by asking her to say that she was his sister. Ramban also says that as "punishment" for going down into Egypt, Avraham's descendants were destined to go down into Egypt. On the other hand... Other sources teach us that we derive halachic permission to leave Eretz Yisrael in the event of severe famine BECAUSE that's what Avraham Avinu did.

Furthermore (still on the same other hand), Pirkei Avot testifies that Avraham Avinu "passed" all his "tests of faith". Being driven out of Eretz Yisrael by the famine and having to subject Sarai to the perils of Paro's Palace are both counted (by some) among the Ten Tests of Faith.

A possible answer might be that how a person behaves in a difficult situation is one thing, and how the situation affects his faith in G-d is another thing. G-d sends someone to a new land, and then hits him with a famine. Regardless of how that someone handles the situation, he might be disillusioned, to say the least. That would mean failing a test of faith. That didn't happen to Avraham. His faith was as strong as ever. Ramban thinks that he erred in judgment as to what he should do. Avraham chose the self-help, HISHTADLUS option. His choice was "approved of" by the subsequent halachic ruling. Maybe he should have chosen the stay-put and trust in G-d option. Ramban thinks he should have. An inadvertent sin, says the Ramban. But no deterioration of Avraham's faith. Those are two different issues.