My chess teacher says that I need to choose either e4 or d4 and stick with it...  I cannot play both.

I feel like I'm choosing a wife...  I can never experiment with other relationships :)

Basically, I'm an attacking player, and I like e4, but d4 and some of the interesting structures also appeal to me...

I am sad that I will be eliminating 50% of the interesting chess positions!!!

Which do I choose?

Well, I don't understand why you have to choose between two of your favorite pawn-thrusts, but you should choose whichever you are better at.

If you can excell at both, wouldn't you be a potentially better player?

EDIT: Thanks, now I realize why what I just said isn't the correct way of looking at it! (no sarcasm intended)

Cannot d4 be tactical and sharp also, if you choose certain variations?  Or am I wrong?

I guess you should just play tactically with 1.d4.

Sure they can. Morozewitch, Tal, Shirov, Kasparov, Topalov etc all played d4. It is the player who is tactical not the opening,

Stick with one for about a year (I recommend 1. e4), then you can learn the other one. 

GMs today tend to know more than one opening to a serious level (where they can play either without jeopardizing their rating). A large amount can play d4 and then e4 at the same level. Or they can play the Classical French, and then the Advance French with similar expectancy of results. 

This is different to years ago when GM repertoires tended to be a lot narrower by comparison. I guess the modern theory goes that by studying different openings you are broadening your chess knowledge significantly more than learning another few moves in a specialized opening. So the energy expended in learning the moves of another system might pay for itself.  

A rare victory in the evolution of modern chess... you don't have to stick to the same old openings or risk wasting your time.   

How long did Kasparov play d4?  Didn't he play it earlier in his career and then switched later?

There are a lot of variations within 1.d4 that are very tactical like in the Semi-Slav 5.Bg5 dxc4 (Botvinnik) or 5.Bg5 h6 6.Bh4 (Anti-Moscow gambit) and 1.d4 d5 2.c4 e6 3.Nc3 c6 4.e4 (Marshall Gambit) or 4.Nf3 dxc4 (Noteboom). Also the Grunfeld Exchange variation with Nf3/Rb1 gambit. The Catalan (yes, the Catalan--its not boring like many ppl make it out to be) can also become very crazy in certain variations.

d4/c4 doesn't always lead to closed positions.

And theres also other lines like 1.d4 2.Nc3 (Veresov) which is more like an e4 opening actually. If you play that you also have to be ready for the CK, French, Benoni, Dutch.

Kasparov has always been a 1.d4 guy but he has played a lot of 1.e4 too. I think he played a lot of Torre Attacks when he was a kid.

you also have Nf3

Once you master one go to the next, the most important is to have fun what ever e4 d4...what ever, your teacher wants you to win, but only you know which one, with which one you win most of the time...that's the one for u!:)

Which one does your teacher play.

It is more logical to learn the e4 openings first.  If you have a good grounding in all those pawn structures, but not the d4 structures, take up d4 until you are familiar with those too.

But if you aren't familiar with the ideas in the various King-pawn openings structures, you will find it easier in the long run to learn them before tackling 1 d4.

I'd have to agree with your teacher. Yes, there are some 1.e4 openings that are closed and positional; and yes, there are 1.d4 openings that are sharp, but by and large e4 is more tactical while d4 is more positional.

There are ways to minimize defense preparation against 1.d4, too. One of my early tricks against 1.d4 as black was to respond 1...d6. If my opponent played 2.e4, we're in a Pirc defense. (Or play 1...e6 for a French, or 1...c6 for a Caro-Kann, etc... anything that allows or encourages 2.e4.) Transposition is your friend, so learn alternate move orders for your favorite openings.

1.d4 d5 2.e4...the Blackmar-Diemer-Gambit is not really positional in my point of view and a good choice for attacking players.

And yet after 2. ...e6 we have a french defense - probably not ideal for someone looking to play the Blackmar-Diemer gambit.

I always play 1.d4 if i know that the black player plays Kings-Indian. I play the Four-Pawns attack. It works for me .

If you're an attacking player, 1.e4 is probably the way to go. However, it doesn't mean you must completely drop 1.d4 out of your practice : you can use it in training games, or simply with a lower frequency in regular games (like once every 5 games or something like that).

Start with 1.e4

and after 3.c4 with have the Diemer-Duhm-Gambit... :-)

or 3.Be3 the Alapin Gambit

Even with 2.Nc3 you can play the Winckelmann-Rainer-Gambit or the Aljechin-Chatard.

I prefer 2.Nc3, because i like to play as white against French...i dont know, why everyone has the opinion, BDG-players dont like to play vs. Caro-Kann or French...

When you play the BDG, you also have to prepare vs. 2...e6 and 2...c6, if not, youre right, the situation starts to get worse.