Having written this, I've just noticed Barrie's answer. By far the easiest thing to do is to learn the constructions, but they can be analysed (as Bill Franke asked in a comment). Whether it's actually worth the effort may be a moot point.
Let's deal with the second sentence first, because that's easier.
The second sentence uses the phrasal verb look forward. While this idiomatic verb means anticipate, it behaves grammatically and requires a preposition to indicate the indirect object — what you are looking at, or in this case, looking forward to (it's "to" because it's concerned with time).
Thus look forward requires to as a preposition and a noun or noun phrase. See is a verb and therefore not right; seeing is a gerund behaving as a noun and is therefore the right form to use.
I look forward to seeing you.
The first sentence has a wrinkle or two. Want is not a simple verb!
Where a verb is followed by a second verb, that second verb is always in an infinitive form. Because want is not a modal verb (such as do or can) that infinitive form uses to.
I can have it [can is never followed by a to infinitive]
I want to have it [want is followed by a to infinitive]
Want can also be followed by a noun, or noun phrase, or a gerund which acts as a noun; that is, it can be an ordinary transitive verb.
I want an apple
That last cake wants eating
In this case, seeing you is not a noun phrase or gerund, it's a present participle and thus ungrammatical. Therefore the next word must be an infinitive form of the verb (see) and because of that it needs to.
I want to see you.
The wrinkle is knowing that seeing presented as a choice here is a present participle and not a gerund like the "look forward" sentence. The clue is that it has its own object, you, which conflicts with the use of want. Want fundamentally indicates a need, a lack, which must be satisfied. As such, it can only have a single object, the one thing which satisfies that need, so you can't use seeing you. Thus you must use see and that must be a to infinitive.
At is used to talk about the position at a point.
Examples:
It's very hot at the center of the Earth.
Turn right at the next traffic light.
Sometimes we use at with a large place when we consider it as a point that exists on a journey, as a meeting-place, or as the place where something happens:
You have to change trains at Didcot.
Let's meet at the station.
In is used for position in a three-dimensional space (when something is surrounded on all sides):
I don’t think he is in his office.
Let's go for a walk in the woods.
Best Answer
This is a very good question and it made me do some serious digging. Here is what I found:
In the examples you have given, it is fine to use a preposition. There are cases when prepositions must be omitted and when omission is optional:
prepositions of time are omitted before the words: last, next, this, that, some, every (We met last month. We meet every day.)
"at", "on", "in" are optional in some cases (but only these three prepositions).
when the phrase refers to times at more than one remove from the present: (on) the day before yesterday, (in) the January before last.
in postmodified phrases containing "the" the preposition is optional in American English: We met the day of the conference., We met the spring of 1983. However: We met in the spring. (can not be omitted because there is nothing after the prepositional phrase.)
in phrases which identify a time before or after a given time in the past or future: (in) the previous spring (the spring before the time in question) (at/on) the following weekend, (on)the next day.
On the whole, the omission is more typical in American English, and normally it makes the phrase less formal.
Source: A Comprehensive Grammar of the English Language, Longman, 1985