The biasvariance tradeoff is a unique result in machine learning: it sits on extremely solid theoretical foundations, and has a ludicrously farreaching scope of applicability. Which might explain why it’s treated so… interestingly… in introductory machine learning courses (or at least, my introductory course). Quickly go over the proof (it’s easy enough that even undergraduates can digest it), build some intuition about what it means for reallife predictors, and move on.
Enough has been said about what the biasvariance tradeoff (a.k.a. the biasvariance decomposition) is (read this excellent essay if you’re shaky). However, not that much has been said about how to manage the bias or variance of your model, which seems to me to be infinitely more important to know. There’s neat advice scattered thinly throughout the interwebs, but I figure I’d consolidate them in one place. I’d also highly recommend these slides from Andrew Ng at Stanford, in which he outlines some practical advice for machine learning models. I basically recreated some of his graphs for two of the three images below.
Of course, take this with a pinch of salt: the most important thing in statistical modelling is not overbearing guidelines or rigid dogma, but the situation you have at hand. Clear thinking is always good!
High Variance
Diagnostics

The testing error looks like it continues to decrease with the training set size, suggesting that more data (specifically, more examples) will help.

There is a significant difference between the training and testing errors.
Remedies

Try to obtain more examples.

Perform some feature selection or feature engineering to get a highquality set of features.

Considering ensembling (e.g. bagging or boosting your predictors).

Consider regularizing your model (or pruning it, or otherwise enforcing model parsimony).

Consider using a less flexible model: perhaps a parametric (or even linear!) model will suffice for your purposes?
High Bias
Diagnostics

Even the training error is unacceptably high.

The training and testing errors quickly converge to a common value.
Remedies

Try to obtain/engineer more features (consider polynomial or interaction terms).

Consider using a more flexible model: perhaps nonlinear or even nonparametric models would work better for your purposes?