What is the best Scheme implementation for working through SICP?

Use MIT Scheme.

It's recommended by the authors of SICP, and is used at MIT for the 6.001: Structure and Interpretation of Computer Programs course.

This was suprisingly annoying to get done on macOS. Here's how it works as of today, assuming you have Homebrew. (Might want to run brew update once in a while).

brew cask install racket
raco setup  # might be optional
raco pkg install sicp

Now you can (require sicp) or simply run

racket -l sicp --repl

Which you might want to abbreviate to scheme. In bash that'd be

alias scheme='racket -l sicp --repl'

which you can add to your ~/.bashrc

Use Racket (formerly PLT Scheme).

The DrRacket IDE is an excellent starting point for all things Scheme including SICP.

To look up keywords in the documentation, place the cursor on the keyword and press F1. In DrRacket you can now see the images directly in the REPL (the read-eval-print-loop).

SICP Support for DrRacket, by Neil van Dyke.

Update (2016): The new SICP package is at http://pkgs.racket-lang.org/#[sicp] Download it with the Package Manager (in DrRacket) or use raco. Manual

Update2 (2016): Also if you want to try a new implementation of the SICP picture language, then download sicp-pict2.rkt.

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Update3 (2020): The sicp-lang package includes an implementation of the sicp language and the sicp picture language.

Documentation: SICP Support for DrRacket
Source Code: sicp-lang on GitHub
Racket Package: sicp-lang package info

Note: The picture language has more features (such as colors and larger sizes) than the picture language described in the book. The source contains examples: https://github.com/sicp-lang/sicp/blob/master/sicp-pict/main.rkt

MIT/GNU Scheme, just make sure you load the SICP compatibility package (yes, they provide specific libraries to enhance guarantee the SICP exercises work).