Guten Tag, Leute,.

Commas are a slippery thing. We Germans use them more often than english people. Even most Germans aren't sure where to put them and how many. If you wanted to go for a traditional letter opening, the period would become a comma, there would have to be an empty line and the words at the beginning of the next line wouldn't be capitalized, because it's the same sentence.
But you only use that in letters and e-mails.

Heute, der fünften Mai, ist der erste Tag, an derm ich meine Abendteuer mit Deutsch imn LangCorrect beginne.

"der fünfte Mai" is a parentheses. Germans love parentheses (see Mark Twain). A parentheses is like an afterthought (Today is the first day. Today, the fifth of May) put into the sentence, therefore it has to be "fenced off" with commas or hyphens.
"fünfte" is just normal declination for "der Mai".
Here, the comma is superfluos, but a lot of Germans probably would have made that mistake, and some would argue about it.
You need the preposition "an" because you start at ("an") this day.
"dem" is Dativ of "der" and needed at this place because:
You start at which day? This day. In German this needs the Dativ of "der Tag" which is "dem Tag".
"Abenteuer" is neutrum, so you need "mein" not "meine", that would be feminine.
"Abenteuer" has nothing to do with the evening ("Abend"), but for me it is curious that a non-native German speaker could come to the conclusion because they sound similar. "Abenteuer" has the same latin root as "adventure" it just changed slightly more in German.
I would use "in" instead of "im". But that is slightly controversial with brand names in general in Germany. You can got "zum Laden" or "in den Laden" and "zur Arbeit" or "auf die Arbeit" or even "in die Arbeit". If you use brand names like Aldi, Rewe, Lidl all that gets even more complicated, but "in" sounds better in this case. If you are interested search for the whole "die Nutella oder das Nutella" debate.
Generally you would avoid "first day" and "beginning" in German because "first" implies it's the beginning of something so it's obsolete.
You could say: "Heute ist mein erster Tag meines Deutschabenteuers mit LangCorrect" but that is a very complex sentence for beginners or: "Heute ist der Tag an dem ich mein Abenteuer mit Deutsch in LangCorrect beginne" just dropping the first day.

Ich hoffe gerne, dass ich mein Deutsch verbessern kann und viele Korrekturen bekomme.

Except for the superfluous "gerne" very good.
If you want to strengthen the "gerne" you have to use "sehr", "ziemlich" or, depending on context, (see, we Germans just love parentheses) "stark".
Very good use of the ", dass" ("Komma dass", very often heard in German) because even some Germans do it wrong consistently.

Herzliche Dank, und viele Grüße

Just a typesetting thing, but you would write them together and in one sentence. Except if you want to make both extra clear and to stand out, but then they have wo be seperate sentences with periods and it would be better to strengthen the "Dank" and "Grüße" with adjectives.