C'mon, guys...it says over 20 people have read this and you mean to tell me no one knows?
Best move against d3/d4?

There are many ways out there... if d3/d4 means closed game... so check out for close games' openings and find which 1 u feel most comfortable with then that is ur personal best move against d3/d4
Not sure if I should be throwing bones like this (I'm a 1.d4 player, albeit not very good lol) but ...d5 or ...Nf6 are the most common responses - usually leading into the Queen's Gambit and King's Indian Defense respectively. Hit opening explorer or wikipedia and go from there, there's a lot of material to look at.

Against 1.d3 play whatever you like to play against e4 (because you can probably guess e4 is comming next or soon), or just stick to playing 1...d5 like anthony said... or just play anything you like... because 1.d3 is nothing, it can turn into many many things and doesn't stop black from playing any first move.
1.d4 is a common opening and you shold have something against it. Like anthony said 1...d5 is a good way to respond. Especially if you're a beginner (although of course grandmasters play this too).
againts d3, uhh d5 yea but d3 aint no problem!!!
Against d4, either d5 or Nf6.
But there is no BEST move..
Since d3 and d4 are two very different ways to start black's repsonse is most likely also different in those two cases. Anayway, what is the best move against d4? That depends on the player, so it is a subjective matter really. Let me explain by some examples:
I sometimes play 1... Nf6 going into an Indian defense, I like King's Indian, sometimes I play 1.. Nf6 followed by 2... c5 going into the Benoni, it can happen that I play 1... c5 right away. If I am in the mood for 1... Nf6 and feel bored by any other move then 1... Nf6 is the best move since I play better when I play a move that awakes my excitement.
Then it can happen that I am fed up with playing 1... Nf6. In that case it is a bad move. Then another move is the best move. Here are the moves that I like to play besides 1... Nf6 and 1... c5:
1... c6 inviting white to play Caro-Kann which he can do by playing 2. e4. Otherwise we will probably go into a variation of Queen's Gambit.
1... d5 and it is up to white whether to play Queen's Gambit or something else.
1... f5 playing Dutch.
Each of those can be "the best move" because of my own mood. Sometimes I play something else but one of those.
But, as has already been said, the moves 1... Nf6 and 1... d5 are very common. They are good moves, there is just no best move objectively speaking.
Regarding the other option 1. d3: It is very rare that I encounter someone playing that move. What I would answer probably depends on what I would like that day.

There are many ways out there... if d3/d4 means closed game...
Not always. Some d4 games are considered open:
Catalan Opening: Open Defense (E04)
1.d4 Nf6 2.c4 e6 3.g3 d5 4.Bg2 dxc4 5.Nf3
5,280 Games in Database
RESULT:
36.14% 37.65% 26.21%
EXPLORE THIS OPENING
POSSIBLE CONTINUATIONS:
move games white win / draw / black win
5...Nc6 777 32.95% 38.87% 28.19%
5...a6 740 34.19% 33.65% 32.16%
5...c5 563 38.01% 44.4%
5...Be7 229 45.41% 36.68%
5...Nbd7 213 47.89% 31.92% 20.19%
5...Bb4 180 36.11% 40.56% 23.33%
5...Bd7 118 38.98% 35.59% 25.42%
5...c6 115 41.74% 22.61% 35.65%
5...b5 72 50% 33.33%
5...Bd6 2 100%
5...b6 1 Veinger Itzhak - Lederman Leon (1978)
5...Na6 1 Horvath Csaba - Shariyazdanov Andrey (1995)
Recently I've noticed a lot of my opponents are playing that move. Is it a good opening? And what's Black's best answer?