An opening for black that’s effective vs e4 and d4

No there isn't, though there are a couple of openings that allow you to develop your pieces in roughly the same way, the Slav/Caro-Kann and Modern Defense being obvious examples.

I don't like 1.d4 c6 for black - why not just play 1...d5 2.c4 c6? 1...c6 is less flexible against the London System and the Jobava-Prie Attack.

A common mistake people make is they think the caro-kann and Slav are the same because Black's pawn structure is the same.  Major problem with that train of thought.  There are 16 pawns, not 8.  There are 32 total pawns and pieces, not 16.

So long story short, NO, there is not!  The Pirc is not the King's Indian.  The e4 modern is not the d4 modern.  The Benoni is not the Sicilian.  The QGD is not the French.  Actually, the QP opening most similar to the French is the King's Indian Defense because of the severely blocked center and bad bishops (light in the French, dark in the King's Indian)

"Play 1...d6 Against Everything" by Zude  & Hickl....

Against 1.e4 it features the Antoshin Variation of the Philidor Defense.  Against 1.d4 it employs the Old Indian Defense.

https://www.amazon.com/Play-Against-Everything-Ready-use/dp/9056917447/ref=sr_1_fkmrnull_1?crid=166BNRC0J50YX&keywords=play+1...d6+against+everything&qid=1552112467&s=books&sprefix=Play+1...%2Caps%2C209&sr=1-1-fkmrnull

https://www.chessable.com/play-1d6-against-everything/course/13376/

https://www.newinchess.com/media/wysiwyg/product_pdf/9051.pdf

"... If you choose the Pirc against 1 e4, it makes sense to consider the King's Indian against 1 d4. This is more flexible and will give you additional options later. … Likewise, the Caro-Kann and the Slav fit together, and then you can answer 1 c4 by 1...c6, without having to undertake any additional learning to cope with 2 e4. …" - GM John Nunn (1998)

"One particular approach to opening repertoire management is the use of universal systems, ... The use of such systems can enable a player to reduce the amount of opening theory he needs to study, and to reach positions of a type he is familiar with and enjoys playing. It is to the pros and cons of this approach that we now turn. ... it is rather more difficult for Black to adopt a universal system, ... The most popular one ... I will discuss in the next section. ... I refer to the King's Indian/Pirc/Modern complex. These lines are characterized by a kingside fianchetto, with such moves as ...g6, ...Bg7, ....d6, ...Nf6, etc. ... The lack of early central contact ... means that there is little chance of violent early contact knocking the player out of his preferred scheme. ..." - FM Steve Giddins (2003)
If I remember correctly, GM Seirawan suggested this sort of thing in Winning Chess Openings.
https://web.archive.org/web/20140627132508/http://www.chesscafe.com/text/hansen173.pdf
Also discussed in A Practical Black Repertoire with Nf6, g6, d6
https://www.newinchess.com/media/wysiwyg/product_pdf/7655.pdf
https://www.newinchess.com/media/wysiwyg/product_pdf/7632.pdf

Perhaps, it would be of interest to look at The Fianchetto Solution by Emmanuel Neiman and Samy Shoker (2016)

https://www.newinchess.com/media/wysiwyg/product_pdf/9029.pdf

Look carefully at available samples before buying. I think that most (perhaps all) of that stuff would not be very helpful for someone with a rating below 1000. Might be a good idea to start with Discovering Chess Openings.
https://web.archive.org/web/20140627114655/http://www.chesscafe.com/text/hansen91.pdf

This (i.e., the book and its openings repertoire) is far simpler than trying to learn the Kings Indian/Pirc/Modern or the Slav/Caro-Kann complexes.....especially for those that don't have "Master" in their chess title (e.g., the OP)...

Against e4 you may think for sicilian defence, but the content of it is vast. 

If it's a  BLITZ match then try Colorado gambit. Learn it from GJ_chess on youtube

The other option is Caro kann defence. I myself play caro kann.It has a solid pawn structure and favours black in endgame.

If not then try out Scandinavian defence, nimzowitch defence....

Or else simply play  1......e5

Against d4 you may try Slav or semi Slav defence. It has a solid pawn structure. But if  2.c4 hasn't been played then you need to play on principles of development and good placement of pieces. 

Kings Indian Defence needs very much preparation because if white player is experienced and has pretty much knowledge against KI defence attack gets defended by white or the initiative of black is vanished or Queen's are exchanged then because of solid pawn structure of saemisch variation White will have good endgame. 

Is the book a good choice for someone below 1000?

"... Looked at objectively, the Antoshin Variation and the Old Indian may be somewhat inferior in quality to the usual main openings, but this small disadvantage, after a certain learning phase, is more than compensated for by a better understanding of the positions that arise. ... for club players with an Elo rating between 1400 and 2200. ..." - Erik Zude (2017)

https://www.newinchess.com/media/wysiwyg/product_pdf/9051.pdf

Again, I suggest the careful examination of the available sample before making a decision to pursue this option.

I repeat. The suggested 1...d6 repertoire is simpler to learn than the altenatives mentioned here, for lower rated improving amateurs.  The book itself focuses primarily on themes and presents only the necessary theory to supplement the exposition of the themes.  So its not a tome on the Philidor or the Old Indian defense.  Just enough theory is presented to get one started competently playing the repertoire.

This is not to suggest that this is the optimum repertoire for anyone, including the OP.  I am simply addressing the OP's stated requirement of one opening (or complex of openings) that is playable and has related structures and themes that might be relatively "easy" to learn.

Is there an answer in there about whether or not the book is a good choice for someone below 1000? I suggest basing a judgment on the Zude comment and/or a personal examination of the sample at:

https://www.newinchess.com/media/wysiwyg/product_pdf/9051.pdf

I suspect that, for his target audience, Seirawan had a somewhat lower rating range in mind when he wrote Winning Chess Openings with its chapter on the Pirc etc. approach.

You have a valid point. 

Upon further reflection the book and its repertoire does assume at least a basic understanding of positional chess concepts and a commensurate level of chess sophistication, knowledge and skill.  For someone rated below 1000, it will likely be a relatively challenging read.  But the same can be said regarding the suggested alternatives.

If I had to recommend a repertoire for those rated below 1000, one that is effective and that they can get started playing relatively quickly, I would suggest considering the opening repertoire books by Vincent Moret, especially his repertoire for White.  With the recognition that at some point at least a basic understanding of positional chess concepts is necessary to effectively learn and implement the themes and plans presented in the books.

https://www.chess.com/blog/RussBell/introduction-to-positional-chess-planning-strategy

One can check my (RLBell) review of Vincent Moret's White opening repertoire book here...

My First Chess Opening Repertoire for White by Vincent Moret....

https://www.amazon.com/First-Chess-Opening-Repertoire-White/dp/9056916335/ref=sr_1_1?crid=38HBC44N2TI0N&keywords=vincent+moret&qid=1555930386&s=books&sprefix=vincent+moret%2Cstripbooks%2C194&sr=1-1-catcorr

The Black opening repertoire has evoked some controversy in the sense that the suggested openings comprising the Black repertoire are more disjoint (when considered as a "system") and more challenging to learn that those of the White repertoire (see the reader reviews).  Nevertheless, (IMO) it does provide some interesting, dynamic chess (definitely not boring), and is very playable, even for the beginner-novice...

My First Chess Opening Repertoire for Black by Vincent Moret...

https://www.amazon.com/First-Chess-Opening-Repertoire-Ready/dp/9056917463/ref=sr_1_2?crid=38HBC44N2TI0N&keywords=vincent+moret&qid=1555930417&s=books&sprefix=vincent+moret%2Cstripbooks%2C194&sr=1-2-catcorr

Again, the presentation of both books is to emphasize themes and plans, to get the reader started playing quickly against similarly rated competition, and not to inundate them with reams of theory and variations.  One can always consult additional resources that deal with the specific variations in more depth as desired.

Here is a quote from page 169 of the Zude/Hickl book:

"... B) 11...♖c8 was the better move. Here are a few sample variations that show that the first player has good attacking chances: 12.♖b1 (12.♗d2 e6 13.f5!?∞) 12...♕b6 (12...e6!? 13.f5↑ with the idea ♗g5) 13.♗d2 (13.♘g5? ♗xe2 14.♘d5? h6 15.♘xb6 axb6-+) 13...♖fe8 Black takes the mate at h7 out of the position. Now it is not so easy to push the white attack forward. …"

And that is only part of the "B" comment. Together with the "A" comment, more than half a page was used. I suspect that, for his target audience, Seirawan had in mind a rating range somewhat lower than 1400-2200 when he wrote Winning Chess Openings with its chapter on the Pirc etc. approach.

https://web.archive.org/web/20140627132508/http://www.chesscafe.com/text/hansen173.pdf

RichardNixon37, was, of course, seeking "an opening for black".

For the last time, I will repeat.  Reading the Zude-Hickl book is a "simpler" (albeit not necessarily  "simple") learning process (for the lower rated improving chess amateur) than tackling the previously suggested alternatives of Kings Indian/Pirc/Modern or Caro-Kan/Slav.  Examination of any books or other resources covering those openings will make it readily apparent.  Enough said.

I know.