Cersei has re-armed the faith; this was likely the single largest mistake she has ever made.
She thought of it as a simple political maneuver: The newly-strengthened Faith had been fairly neutral from a political point of view, and Cersei meant to be the first to secure their allegiance - especially against House Tyrell - by making some concessions. Having done so, she immediately went on to use her new toy weapon against House Tyrell.
The faith had been disarmed centuries ago by King Jaehaeris I, after a seven-year war called the Faith Militant Uprising, which was started when King Aenys I slighted the Faith by marrying his son to his daughter. It saw the death of both Aenys I and Maegor I, who both were not able to defeat the Faith Militant and the Lords who had taken up their cause - even though the Targaryens had dragons back then. King Jaehaeris I only disarmed them after giving a full amnesty to all rebels, and promising protection to the Faith.
Now Cersei stands publicly accused, by the Faith, of
- Killing her husband and king, Robert I
- Commiting adultery with her brother Jaime
King Tommen I's claim to the throne would become near-invalid if
Cersei is found guilty.
Above all, Cersei wants to protect her children, King Tommen I and Princess Myrcella, from the fate that was predicted by Maggy the Frog: Cersei will see all of them dead.
After the the fulfillment of many of Maggy's predictions, most importantly the death of King Joffrey I, she has become terrified and tries everything in her power to save her children from their fate and its agent: the "new queen, younger and more beautiful, who will cast her down and take everything she holds dear". This explains her extreme hostility against Lady Sansa Lannister and Queen Margaery Tyrell.
In the Books, The Valonquar (Valyrian: Little Brother) is an other agent of the Prophecy, predicted to kill Cersei. This explains her hatred of Tyrion Lannister (though she has an other little brother, Jaime Lannister).
In addition to protecting her children, Cersei wants power. Both for herself, and for her children. Probably because she sees power as an other means of protecting her children; clearly she should take a look at some statistics...
Cersei's possible course of actions
Attacking the Faith
This is the course suggested by you. She threatened to do this while locked in her cell, but it is entirely unfeasible. After her threats, and knowing her, the Faith will surely expect assassination attempts, so she'll need to do a right-out attack.
In King's Landing, the Faith's forces (The Faith Militant) greatly outnumber Cersei's (The Kingsguard, the Lannister household guard and the City watch). If Cersei calls a Lannister army, it will take weeks to arrive from the Westerlands. Many would flock to the Faith's side: King's Landing's smallfolk would most certainly oppose her (after her walk of shame), and many lords will join for political reasons or due to their pious nature. Most notably, from Cersei's paranoid point of view, House Tyrell might join the Faith in return for the release of Queen Margaery and Ser Loras, and House Martell because they're just waiting for a reason to fight.
In reality, any of the troups of the Reach (for the reasons stated above), the Vale (after Lord Baelish is done with the Boltons), Dorne (because who knows about the true intentions of Prince Doran Martell), the Riverlands (because Lord Walder Frey is hard to predict and will generally side with the winners), and the Stormlands (in revenge for the deaths of King Robert I, possibly to install one of his noble bastards like Edric Storm or a distant cousin) might join the Faith.
I can't think of anybody who would take up arms against the Faith, for Cersei - possibly not even all of the Westerlands would rise.
The conflict probably wouldn't last very long, and most probably end in her and King Tommen I's death at the hands of an angry mob. The victors might hold a Great Council, presided over by the High Septon, to choose a new king (but this is just pure speculation).
Barricading herself in the Red Keep and "waiting it out"
Refusing to stand trial would, in the eyes of most, prove her guilty. The Faith would hold her trial in her absence, and sentence her to die. An angry mob might storm the Red Keep to "carry out her sentence", probably killing King Tommen I as well as her.
Fleeing in the shadow of the night, like Lady Sansa Lannister
This might end up quite well for her, with regards to her primary goal. However, she would have to take King Tommen I with her and abandon all of her hopes for power (her secondary goals).
Her best option would be to flee to Casterly Rock, where she would be protected by house Lannister's bannermen. Like above, she would be senteced to die in her absence. She would bring shame upon her house, but she has shown that she doesn't care about that. King Tommen I's successor might decide to declare war on the Westerlands to bring her to justice, in which case her family or bannermen might deliver her and Tommen to prevent a war. Tommen would be safe, probably as a hostage of the new king, renamed "Tommen Waters".
Given the fact that she is, in fact, guilty of the charges, and there is overwhelming evidence, she would almost certainly be found guilty, and probably sentenced to die. Depending on who is chosen as the new king (Stannis would be an obvious choice, if only he were a bit more alive) King Tommen I would be tried as a traitor, but possibly found innocent due to his young age. In that case, he would remain as a hostage of the new king, renamed "Tommen Waters".
Standing trial by combat
If she loses the trial, the results will be like above. If she wins the trial, she is innocent by law. She will still have lost a lot of influence after her walk of shame, but King Tommen I is safe, and his legitimacy has been proven once and for all. This is the best option she can hope for, securing both her primary and secondary goals.
Only a man of the Kingsguard can stand champion for a member of the royal family, and, after Ser Barristan Selmey was dismissed and Ser Jaime Lannister lost his sword hand, all men of the Kingsguard were rather uncapable. That is, until mysterious Ser Robert Strong joined, just at the nick of time.
Obviously a zombified Ser Gregor Clegane.
She has never seen Ser Robert Strong fight, but he's her best option. I mean, whom will the Fate name to face him? Lancel Lannister?
well, Sandor Clegane, as speculated in other answers, but Cersei doesn't know that.
Can Cersei even choose for herself?
Cersei, not having been Queen Regent for quite some time, had two remaining sources of power:
- The many friends she surrounded herself with, most of them dishonest flatterers.
- The fact that "King Tommen I tasked her to rule for him".
Given her loss of face due to her walk of shame, most of her "friends" will have abandoned her, and given that Lord Kevan Lannister has been back in Kings Landing for quite some time, he surely has established firm control over King Tommen I, for the good of both the king and the realm.
It would be a shame if something happened to him.
It's likely that Lord Kevan will put her under constant watch to prevent her from doing anything that would further dishonor or weaken King Tommen I or House Lannister, or insult their allies of House Tyrell.
What Cersei should do (IMO)
If Cersei truly values the life of her children above all, and fears the prophecy, she should kill herself.
That way she would make certain that she can't see her children dead, making it impossible for the prophecy to come true.
If her death is not revealed to be suicide, but instead can be blamed on, say, House Tyrell (just because) or House Martell (those guys are sneaky poisoners), she'll likely be remembered an innocent victim of plotting, and her son will stay King Tommen I (fulfilling her secondary goal).