Is there a voting method with a sane strategy? 
Is there a voting method where the best strategy for strategic voters can be explained in a sane way?

According to Gibbard–Satterthwaite, there is no "strategy-free" (and reasonable) voting method.  That is, if "honesty is the best policy" for a voting method, then the voting method must ignore the voters or be non-deterministic.
Strategic voting in plurality is often pretty simple: amongst those candidates that have a chance of winning, vote for your favorite.  So vote nearly honestly, but generally avoid third party candidates.
However, violations of the monotonicity criterion and participation criterion are pretty irritating for describing a good "strategy" for lying on the ballot.  In particular, you can cause a winner to lose by voting for them, and you can cause a loser (that you would have voted for) to win by not voting.  In the presence of these "if you try to help, you can hurt" conditions, it seems almost impossible to formulate the winning strategy for a strategic voter.
On the other hand, some fairness criteria do not seem tuned to making strategies easy, so perhaps those criteria and the associated impossibility theorems could be ignored.

Is there a voting method where the best strategy for strategic voters can be explained in a sane way?

I assume there is no such strategy for plurality with elimination, but perhaps I am wrong and am just distracted by monotonicity.
 A: Part of the issue here is what "level" of information you might have available about how other voters are going to vote. A very informative book about manipulation of voting systems is Social Choice and the Mathematics of Manipulation by Alan Tayler, Cambridge U. Press, 2005.
A: If there were a voting system $V$ with a strategic map $S$ from a voter's true preferences to the reliable best way of voting in $V$ given those preferences, then you would have a strategy-proof voting system: just take in all the incoming votes, apply $S$ to them, and do $V$ on the result!
So Gibbard-Satterthwaite proves that there can't be a single canonical strategy, no matter how complicated, that doesn't depend on other voters' behavior.
A: You might want to check out range voting. Range voting is like approval voting, where You can vote for as many Candidates as You like except, while in approval voting You say, "I approve or disapprove of these Candidates", under range voting You say, "I give these Candidate X out of 10 votes, those Candidate Y out 10, and this Candidate Z out of 10 votes." (Note: Range voting is note like simply having multiple votes where You are given, for example, 10 votes and told to allocate them amongst different Candidates. Instead, range voting is more like a movie rating system: 4 out of 5 stars, for example.) Range voting lets People vote for Whom They like while encouraging Them to vote honestly.
