اليوم أذهب إلى المستشفى بالحافلة.

In Modern Standard Arabic, the present-tense verb form (أذهب) can cover several English ideas depending on context: I go, I am going, or I will go.
Here, اليوم (today) strongly suggests a specific time, so in natural English it often comes out as I’m going to the hospital today (or Today I go to the hospital in a more literal rendering). Arabic doesn’t require a separate “continuous” form like English does.

The أ- is the 1st person singular prefix in the imperfect (non-past) conjugation. It marks I.
So أذهب = I go / I am going. (The base verb is ذهب “to go”.)

Yes, you can move it. اليوم is an adverb of time, and Arabic is flexible about where you place time expressions. Common options include:

• اليوم أذهب إلى المستشفى بالحافلة. (Today, I go…)
• أذهب اليوم إلى المستشفى بالحافلة. (I go today…) Both are correct; putting اليوم first gives it extra emphasis.

إلى means to / toward and is used for destinations.
So أذهب إلى المستشفى = I go to the hospital.
It typically indicates movement ending at a place.

الـ is the Arabic definite article meaning the.

• المستشفى = the hospital
• الحافلة = the bus
  Unlike English the, الـ attaches directly to the noun.

The final ى (called ألف مقصورة) represents a long ā sound, like an alif but written differently.
So المستشفى is pronounced roughly al-mustashfā (ending in -ā).

It is actually بِـ + الحافلة (by + the bus) but in writing they join: بالحافلة.
Also, when بِـ comes before الـ, the alif of الـ is not pronounced (and is often called a “connecting” alif), so it flows as bil-ḥāfilati.

Here بِـ expresses means of transportation: by.
So بالحافلة = by bus.
Depending on context, بِـ can also mean with or by means of, but with vehicles it’s usually by.

ـة is تاء مربوطة. In pause (when you stop), it’s usually pronounced like -a / -ah: ḥāfila(h).
In connected speech with case endings (in fully vowelled MSA), it can be pronounced as t (e.g., ḥāfilati), because the grammatical ending attaches.

In most real-life writing, short vowels/case endings are omitted, so you see:

• اليوم أذهب إلى المستشفى بالحافلة. If it were fully vowelled in formal MSA, you might see something like:
• اليَوْمَ أَذْهَبُ إِلَى الْمُسْتَشْفَى بِالْحَافِلَةِ Learners should recognize that the sentence is correct without vowels; case endings are mainly for formal reading/recitation and advanced accuracy.

Arabic typically doesn’t use a present-tense copula like am/is/are. The verb form itself carries tense/person information.
So أذهب already includes I and a non-past meaning; the idea of am going is inferred from context.