I have finally gotten through all the grammar, but I am confused with certain words. For example, I learned about должен, можно, and нужно. But, words like видно and замечательно are confusing me. Like, what does "Я видно" and "Он замечательно" mean, and why is it "Мне нужно", but not "Я нужно".
There are many impersonal constructions in Russian that take Dative. Here you can check some of the impersonal sentences with Dative. Those constructions can be literally translated as It is needed to me (мне нужно), it is wanted to me (мне хочется) or it is liked to me (мне нравится). In English, you have a subject 'I' and verbs that are in active voice. In Russian, it works differently: 'I' is rather passive, so things happen to "I", instead of 'I' being an active subject.
Мне видно - It is visible to me. "Я видно" is incorrect.
Он живёт / чувствует себя замечательно - he is fine (alternatively, you can say, у него всё замечательно)
Hope this helps.
"Я видно" is in itself meaningless. "Я, видно,.." is okay and roughly neans "looks like I'm..." - no actual looking is involved.
Замечательно can work both ways. "Он замечательно прыгает" - he jumps magnificently. "Ему замечательно повезло" - "he got really lucky". Note how you can't say "he really lucky".
"Мне нужно сделать" but "Я должен сделать" - I guess it's just how it is.
Unlike English, Russian can have adverbs as predicatives. I agree with @alamar about "Я видно" и "Он замечательно", but as for "Мне нужно" I can explain it.
"Нужно" means "needed", and it is a whole clause by itself meaning "[it is] needed", both "it" and "is" are omitted in Russian. "Мне" is "for me", so "Мне нужно" is translated word-for-word "for me it is needed", which actually mean "I need", that is how this phrase is working in Russian.
In Russian, "I need..." is literally expressed as "to me is needed..." = мне нужно and "I like..." is literally expressed as "to me is liked..." = мне нравится. This is why, for instance, "I need the car" = Мне нужна машина with машина in the nominative case. Although in English the word car is the object, so you may think it belongs in the accusative case in Russian, the Russian construction is saying "To me is needed a car" or, a bit less awkwardly, "To me a car is needed." Either way, car is not the object of the sentence but the subject, hence nominative. Similarly, "I like geometry" is мне нравится геометрия, with geometry in the nominative case since the Russian construction is "To me is liked geometry" or "To me geometry us liked."
Using a more direct construction can have a different meaning than you may intend. For example, "he is cold" is expressed as ему холодно = "to him it is cold." If you instead use the adjective холодный to say someone is cold (он холодный человек or он холоден, using a short form adjective in the second version) then that means the person is cold as an emotional state.
Impersonal sentences have a so-called logical subject. You can interpret this as that the actor undergoes some kind of action or is forced one way or another to perform the action. The suggestion is that the active free will of the actor is absent: rather, the person undergoing the action has no choice.
The actor is forced to perform an action:
Всё-таки Сергею придётся рассказать всю правду. (Sergey needs to tell the whole truth)
The actor doesn't know or cannot decide whether the performance of the action is necessary or possible:
Кому из нас ехать? (Who of us should drive?)
Мне ли тебе рассказывать, как раньше жилось?! (Do I need to tell you, how we lived before?!)
The actor doesn't feel like performing the action:
Сходи в магазин, если тебе не лень. (Go to the store, if you it's not to hard for you. [litt. "if you are not too lazy"])
The actor is not capable of performing the action, because something or somebody is absent:
Нам некогда здесь торчать. (We don't have time to hang around here)
The actor is in some kind of emotional or physical state:
Мне холодно! (I am cold!)
Мне весело на вечеринке. (I'm having fun at the party.)
The actor is (not) in the mood for something:
Татьяне не спалось, не читалось. (Tatyana couldn't sleep nor read.)
The actor is undergoing some psychological activity that is not controlled by her:
Всем казалось, что ты разумнее. (Everybody thought that you were more reasonable.)
The actor is in a physical state that is hardly controllable:
Ему икается. (He has the hiccups.)
So now you know a little about impersonal sentences. Think of them as some action that is happening to a person, rather than the person actively performing the action himself. This is just a peculiarity of the language that you need to grasp, but from personal experience I know it'll become second nature eventually (
просто тебе нужно время ;-)
You ask why you can't say
я нужно instead of
мне нужно. Well, that's because
нужен (нужна, -о, -ы) changes according to the grammatical subject of the sentence. And since we've just seen that the logical subject of the sentence takes the dative, we must conclude that (unless you are a
дерево and want to say something like
я нужно человечеству because you produce oxygen) this is not a correct sentence.
Compare the following sentences and see how the logical subject is different from the grammatical subject:
Мне нужен ты! (I need you!)
Ему нужна девушка (He needs a girlfriend)
Нам нужно работать (We need to work)
Им нужны деньги (They need money)
 Examples are taken from Honselaar & Podgaevskaja (2007), "Учебная грамматика русского языка", Pegasus, Amsterdam.