أريد أن أبدأ الدرس العاشر اليوم.

In Modern Standard Arabic, verbs like أريد (I want), أحب (I like), أستطيع (I can), etc. commonly take a following verb in the structure أن + فعل مضارع (literally to + present verb), e.g. أريد أن أبدأ = I want to start.
You can sometimes omit أن in some styles, but the standard, most “textbook” MSA pattern is to include it.

Yes. أريدُ is فعل مضارع (imperfect/present) in the 1st person singular: I want.
It comes from the root ر-و-د (form IV: أرادَ / يُريدُ).

Yes, أبدأ begins with همزة on an ألف (أ). It’s pronounced with a clear glottal stop at the start: ’abda’u (approx.).
The verb is from بدأ (to begin/start). In MSA: بدأَ / يبدأُ. Here it appears as أبدأُ = I start / I begin (1st person singular).

After أن, the following imperfect verb is in the subjunctive (المضارع المنصوب).
So with full case endings, it would be: أريدُ أن أبدأَ ... (ending -a).
In most everyday unvowelled writing you won’t see the -َ ending, but grammatically it’s there.

A fully vowelled version (one correct possibility) is:
أُريدُ أن أَبدَأَ الدرسَ العاشرَ اليومَ.

• أريدُ ends in -u (indicative)
• أبدأَ ends in -a after أن (subjunctive)
• الدرسَ is often object (مفعول به) → accusative
• العاشرَ matches الدرسَ in case/definiteness
• اليومَ is commonly treated as an adverbial accusative of time

(You may also see slight variations depending on analysis and style.)

It’s not an iḍāfa. It’s a noun + adjective phrase:

• الدرس = the lesson (noun)
• العاشر = the tenth (adjective/ordinal) describing الدرس
  Because it’s an adjective structure, both words take الـ: الدرس العاشر.

(If it were iḍāfa, you would not normally put الـ on the second word in the same way.)

Arabic adjectives must match the noun in definiteness. Since الدرس is definite (the lesson), the adjective must also be definite: العاشر.
If you made the noun indefinite, you could say: درسٌ عاشرٌ = a tenth lesson (less common in meaning, but grammatically valid).

عشرة is the cardinal number ten.
العاشر is the ordinal tenth.
Arabic uses the ordinal form to say “Lesson 10”: الدرس العاشر (the tenth lesson).

اليوم (today) is a time expression and is fairly flexible in position. All of these can work in MSA (with slight emphasis differences):

• أريد أن أبدأ الدرس العاشر اليوم. (neutral)
• أريد أن أبدأ اليوم الدرس العاشر. (emphasizes today)
• اليوم أريد أن أبدأ الدرس العاشر. (fronting today)

Arabic often uses VSO, but it also commonly uses SVO and other orders depending on style and what you emphasize.
Here you have a verb-initial clause: أريد (verb + implied subject “I”). The subject pronoun أنا is not needed because it’s already built into the verb form. You can add أنا for emphasis: أنا أريد أن أبدأ...

Replace اليوم with غدًا (tomorrow):
أريد أن أبدأ الدرس العاشر غدًا.
With full endings: غدًا is typically written with tanwīn fatḥ (ـًا) because it’s usually indefinite and adverbial.

You usually don’t need سـ or سوف because أريد أن... already implies intention toward the future.
If you specifically want “I want to start (later / will start)”, you could say:

• أريد أن أبدأ الدرس العاشر اليوم. (already fine)
• أريد أن أبدأ الدرس العاشر اليومَ (with endings)
  Using سوف would typically go with a direct future statement (سوف أبدأ...) rather than after أن in this exact structure.

A rough pronunciation (without full case endings) is:
urīdu ʾan ʾabdaʾa ad-darsa al-ʿāshira al-yawm
Key points:

• Clear hamza at the start of أريد and أبدأ
• الدرس is pronounced ad-dars (the l of الـ assimilates to د because د is a “sun letter”)
• العاشر keeps the l sound: al-ʿāshir (because ع is a “moon letter”)