"I will have a party today."
You technically can't just add します to every noun there is to make it a verb. You could and people would understand what you were trying to say, but it would sound really weird for some words.
Generally if a noun is listed as a "suru verb" in the dictionary, then it's normal to attach する to it to make it into a verb.
を is used in 「パーティをします」to indicate the specific thing to the verb し (To do), meaning "I do a party" べんきょ is the verb "Study" so you don't need that to say "I study", only if you are saying something like "I am studying spanish" (スペイ語をべんきょします） を Is only used to indicate the specific thing done in the verb ;*~ I hope I'm being understandable…
just a tiny correction: べんきょう is not a verb meaning "to study", but rather a noun meaning "the activity or process of studying".
On that note べんきょうis what is called a suru noun, because you can easily add する (polite form: します) to the end of it to turn it into a verb. Keep in mind not all nouns are suru nouns (though you probably already figured that out from パーティー). From what I've heard they're mostly 2-kanji words originally from Chinese that qualify.
It's two alternative ways of writing the same thing; either one is fine for all します verbs. If there is another を in the sentence, you have to use the form without を however. There can't be two of that particle in the same sentence.
But are you emphasizing the day or the party? In English you might emphasize the day when saying, "today I will finally finish my graduate work". The day is significant as in "today is THE day". But if you're just saying the word "today", there's no need for emphasis everytime. Remember, less is more in Japanese. So yes, you could say は and it wouldn't be wrong, but is that what you want to emphasize in this sentence? There may be context with this sentence where you would. Such as in a long awaited party. Hope that helps.
I'm actually confused as to why the は particle is used here. I know it's a topic marker but "today" isn't the topic I wouldn't think.
Also, if you WERE going to add 私 (though I realize it's unnecessary) where would it go? would it be,
「私今日はパーティーをします」? or 「今日私はパーティーをします」?
Party is the direct object - the thing that is "receiving" the verb action ("to do" in this case). It needs to be marked with a を.
Besides, what particle would you use with the 今日 then? 今日に is incorrect if that's what you were thinking. You don't use に with 今日 except in 1 very specific exception (which is not the case here and is fairly advanced and hasn't been covered).