فترة التجربة تبدأ اليوم.

The subject is فترة التجربة.

More literally, this is trial period or period of the trial as one unit. The verb تبدأ then tells you what that subject does: begins / starts.

So the structure is:

فترة التجربة = the trial period
تبدأ = begins
اليوم = today

Because فترة التجربة is an iḍāfa construction, often called a construct phrase.

In an iḍāfa:

• the first noun does not take الـ
• the second noun can make the whole phrase definite

So:

• فترة = period
• التجربة = the trial / the experiment

Together:

• فترة التجربة = the trial period

You do not say الفترة التجربة here.

Yes. That is exactly the idea.

Arabic often expresses X of Y by simply putting two nouns next to each other:

• فترة التجربة = period of the trial
• naturally translated as the trial period

This is one of the most important noun patterns in Arabic.

Because the subject فترة is a feminine singular noun.

In the present tense, Arabic verbs agree with their subject. Since فترة is feminine singular, the verb is also feminine singular:

• تبدأ = she/it begins

Here it does not mean she. It means it begins, referring to فترة.

Yes. That is also correct.

Both are natural:

• فترة التجربة تبدأ اليوم
• تبدأ فترة التجربة اليوم

The difference is mainly one of style or emphasis.

Very roughly:

• فترة التجربة تبدأ اليوم puts early focus on the trial period
• تبدأ فترة التجربة اليوم starts with the action begins

Arabic allows this flexibility more than English does.

It is usually analyzed as a nominal sentence because it begins with a noun phrase:

• فترة التجربة = topic / subject
• تبدأ اليوم = predicate

Even though the predicate contains a verb, the sentence starts with a noun, so learners are often taught to view it as a nominal sentence.

If you changed it to تبدأ فترة التجربة اليوم, then it would be a more clearly verbal sentence.

Because Arabic does not normally use a present-tense copula like is/am/are in simple present statements.

But note: this sentence already has a real verb, تبدأ = begins, so English is is not needed anyway.

Arabic is simply saying:

The trial period begins today.

Not:

The trial period is begins today.

By itself, اليوم can mean:

• today
• the day, depending on context

In this sentence, it functions as an adverb of time, so it means:

• today

So تبدأ اليوم = begins today.

In fully vocalized MSA, it would typically be:

فَتْرَةُ التَّجْرِبَةِ تَبْدَأُ اليَوْمَ

The endings are:

• فترةُ: nominative, because it is the subject
• التجربةِ: genitive, because it is the second term of the iḍāfa
• تبدأُ: present indicative
• اليومَ: accusative as an adverbial expression of time

In normal everyday writing, these short vowel endings are usually omitted.

The ل of الـ is not pronounced here, because ت is a sun letter.

So:

• written: التجربة
• pronounced roughly: at-tajriba

The t sound is doubled.

So a rough pronunciation of the whole sentence is:

fatrat at-tajriba tabda'u al-yawm

If you include full case endings more carefully, it is closer to:

fatratu t-tajribati tabda'u al-yawma

It is pronounced roughly:

tabda'u

Breakdown:

• تَ = ta
• بْ = b
• دَ = da
• أُ = 'u

The last part contains a hamza sound, so it is not just tabda. There is a slight catch in the throat before the final vowel:

tab-da-'u

This comes from the root ب د أ, which is related to beginning.

Because Arabic present-tense verbs usually include the subject inside the verb form.

So تبدأ already means something like:

• she begins
• it begins

Since the subject here is فترة and that noun is feminine singular, the natural translation is:

• it begins

English needs the separate subject plus the verb form begins, but Arabic builds that agreement into the verb.

Yes, depending on context.

The noun تجربة can refer to things like:

• a trial
• a test
• an experiment
• an experience, in other contexts

So فترة التجربة could be translated as:

• trial period
• testing period
• probationary period, in some contexts

The exact English wording depends on what kind of تجربة is meant.