Do I have to say "A'udhu billah" before reading the Quran?

No, most scholars say it isn't strictly required, but it's a strongly recommended sunnah. Allah tells us to seek refuge from Satan before reciting, so you say "A'udhu billahi minash-shaytanir-rajim" once when you start. If you forget, your recitation is still valid and rewarded.

Where it comes from

The instruction is right in the Quran:

فَإِذَا قَرَأْتَ ٱلْقُرْءَانَ فَٱسْتَعِذْ بِٱللَّهِ مِنَ ٱلشَّيْطَٰنِ ٱلرَّجِيمِ "So when you recite the Qur'an, [first] seek refuge in Allah from Satan, the expelled [from His mercy]." Quran 16:98

Most scholars from all four schools (Hanafi, Maliki, Shafi'i and Hanbali) understand this command as a recommendation, not an obligation. A minority of early scholars held that it is obligatory. Either way, it's an easy habit worth keeping.

What exactly to say

The common wording: "A'udhu billahi minash-shaytanir-rajim", meaning "I seek refuge in Allah from Satan, the outcast."
When: once, at the start of a reading session. You don't repeat it before every verse.
Out loud or quietly: either is fine. Many people say it softly even when they then recite aloud.

Seeking refuge with these words isn't only for the Quran. When the Prophet ﷺ saw a man red with anger, he said the man's anger would go if he said: "I seek Refuge with Allah from Satan" (Sahih al-Bukhari 3282).

What about "Bismillah"?

Bismillah (in the name of Allah, the Most Gracious, the Most Merciful) is a separate thing:

At the start of a surah: say it after A'udhu billah, for every surah except Surah at-Tawbah (chapter 9), which has no Bismillah.
In the middle of a surah: A'udhu billah is enough. Saying Bismillah too is allowed, and many reciters do.

Inside the prayer

In prayer, the schools differ slightly. The Hanafi, Shafi'i and Hanbali schools recommend saying it quietly before al-Fatiha, with differences over whether it's said only in the first unit or in every unit. The Maliki school generally doesn't say it in the obligatory prayers. All of these are valid; follow what you were taught or what your local imam does.

If you forget

No need to restart. If you remember partway through, you can say it then and carry on. Don't let it become a source of anxiety: it's a protection and a blessing, not a password. For more beginner basics, see Where should I start reading the Quran as a new Muslim? and How much Quran should I read every day?

Key points

Saying A'udhu billah before reciting is a strong sunnah.
Most scholars say it's recommended, not obligatory.
Add Bismillah at the start of every surah except at-Tawbah.
Forgetting it doesn't spoil your recitation.

Tip: Make it automatic by tying it to opening your Quran or app: open, say A'udhu billah, say Bismillah, then begin. After a week you won't have to think about it.