Learn Arabic (Modern Standard): سأزور الشاطئ غدا صباحا.

The prefix سـ (sa-) means "will" and attaches directly to the beginning of a present tense verb. Here, it attaches to أزور (I visit). Because أزور starts with an alif (أ), they combine into سأ (sa-'a), making سأزور (sa-'azūru - I will visit).

Could I use the separate word سوف instead of the prefix سـ in this sentence?

Yes! You can say سوف أزور (sawfa 'azūru) instead of سأزور. Both mean "I will visit." The prefix سـ is just a slightly shorter, more common way to express the near future.

Why do the words for "tomorrow" (غدا) and "morning" (صباحا) both end with an extra alif and two small dashes?

That ending is called tanween (specifically tanween fath), which is pronounced "-an". In Arabic, words that tell us when something happens (adverbs of time) usually take this ending to show their grammatical role. So, غد (tomorrow) becomes غدا (ghadan) and صباح (morning) becomes صباحا (ṣabāḥan).

What is the final letter in الشاطئ (the beach) and why does it look like a dotless yaa?

That is a hamza sitting on a "seat" shaped like a dotless yaa (ئ). In Arabic spelling, a hamza at the end of a word chooses its seat based on the short vowel right before it. Since the letter before it (ط) has a short "i" vowel (kasra), the hamza is forced to sit on the matching "i" shaped seat.