Your suspicion is a good one. There are two issues to address: what calculation will achieve the desired result and how to do it. The "how" is answered by a large, flexible array of choices, including the Raster Calculator, as well as the command line and Python scripting. Which you use depends on many contextual details not mentioned so far, so let's skip to the "what."
A good starting point for figuring out the "what" of raster calculations is the Spatial Analyst help system. Once you are familiar with its organization--which is partly in terms of the purpose and partly in terms of the nature of the calculation--you will find your choices are considerably narrowed. Here, you would look first at the math toolset, the reclass toolset, and possibly the local toolset. As in many situations, you won't find exactly what you need, but you will find the ingredients of a solution--usually of many solutions.
(What you need would be well served with a signum function, but it appears this basic tool has been omitted from SA.)
Many people would be satisfied with a reclass command or some multiple invocation of con, either of which would serve them well. However, I find that many forms of reclassification can be expressed efficiently and elegantly with mathematical formulas, so I will offer one here. It relies on the convention that Boolean "false" values are represented as zeros and "true" values as ones. So, for a hypothetical raster named "grid", consider the effect of this expression:
("grid" > 0.001) - ("grid" < -0.001)
The presence of two distinct numbers 0.001 and -0.001 causes us to consider three mutually exclusive, exhaustive cases:
A value truly exceeds 0.001. In this case the first term,
("grid" > 0.001), is true (equal to 1) and the second is false (equal to 0). Their difference is 1, as intended.
A value lies between -0.001 and 0.001, inclusive. In this case both terms are false (equal to 0), so their difference is 0, as intended.
A value lies less than -0.001. The terms now equal 0 and 1, in that order. Their difference is -1, as intended.
The syntactic details depend on which version of Spatial Analyst is used: subtle differences in syntax processing exist between SA 9 and SA 10 and even within intermediate releases. Liberal use of parentheses is usually advisable and some care may be needed with the placement of white space and quotation marks. Experiment on small grids first until you are sure the syntax is right, check the results, and only then perform the operation for real.