Quote Originally Posted by kgcole View Post
I do have a question about the genitive case, though.
OK, and what is you question? I cannot see it.

Quote Originally Posted by kgcole View Post
I’m familiar with the phrase “У меня есть”, but I have read it where “есть” is omitted and it’s written in short as simply “У меня…”; i.e. “У меня книгу”. Is this correct to do this?
You cannot say "У меня книгу". You can say "У меня книга". Note that "книга" is grammatically a subject here! (Lit.: At me is a book).

If your question is about the difference between "У меня есть книга" and "У меня книга", just remember that "У меня есть книга" is better.
I can try to explain some rules where to use "есть" and where to omit it, but I'm afraid it can be a bit complicated for the beginner.

Basically, we always omit "есть" when something naturally belongs to you: "У меня длинные волосы" (I have long hair) - you cannot put "есть" here, since the hair belong to you naturally. Otherwise, if you say "У меня есть длинные волосы" would sound like you have some long hair separate from your body (e.g. you keep that hair in a box or something like that).

Quote Originally Posted by kgcole View Post
I am a little confused over certain articles such as “тот”, especially when it’s declined as “тут”.
1. "тот" is not an article. We just do not have articles. "тот" is an indicative pronoun.
2. "тут" means "here", it's almost the same as "здесь" (if you know). It has nothing to do with "тот".
3. Yes, "тот" IS declined: тот, то, того, тому, тем, (о) том, та, той, той, ту, той, (о) той, те, тех, тем, теми, (о) тех.

If you have a question about the declension, just provide an example and ask what is not clear in that example. We'll try to help.