Finding Rare Birds
Some thoughts on trying to find a rare bird, from someone who has spent an unreasonable amount of time doing just that...
Chance favours the prepared. Having ID knowledge of both common and unexpected bird species is an asset. An awareness of potential vagrant occurrence also improves your chances (i.e., understanding established patterns of vagrancy).
When vagrant hunting you should also find and expect to find aberrant plumages of common birds, rare subspecies, early or late migrants, escaped exotics, etc.
Photograph birds that catch your eye (all species) for future study and learning.
Consider a "detectability bias" when learning about patterns of vagrancy. Cryptic rarities are likely under-detected. (e.g., hard-to-identify birds like gulls or hard-to-locate birds like longspurs). More flashy or easily identified birds may be harder to "find" on your own (e.g., known Varied Thrush records are likely high due to their bright and distinctive colouration, combined with their frequent appearance at bird feeders where they make frequent visits to get noticed and photographed).
Birding at concentration points may give a greater return on your effort (i.e., volume). Visiting locations with a diversity of habitats can further improve your chances.
Finding ways to get better looks at birds (i.e., closer and longer views, ideal light, shelter from wind and rain) increases your chances of spotting AND identifying/documenting an unusual bird.
Be prepared for slow days. By nature, rare birds may be going against the grain, so your best odds of finding vagrants may bring you to locations with low activity for the current season.
Consider checking areas with low observer coverage.
Expand your search radius in the field. Use lower powered optics (e.g., 8x binoculars vs. 10x) for a wider scan. Don't just focus on one area (e.g., the horizon) when birds may be up high or right on shore. Also, try not to skip identifying passing birds as a vagrant may be on the fringe of a flock.
Incorporate an understanding of how weather affects bird migration and vagrancy to enhance many of the above points.
Keep an open mind. Learn about difficult identifications from the perspective of "why not" as well as "why". (Just don't report your "why not a ..." to the local bird listers or they may get upset with your active learning process).
While each of these points are worth considering as you hunt for rare birds, there are two KEY considerations which improve the chances to anyone looking for vagrants. They are:
Go birding as much as you reasonably can. The more frequently you go birding and the longer you spend in the field (specifically looking at birds, driving doesn't count!) is the best way to improve your chances of finding rare birds. Ideally this comes in the form of a local patch! But also:
Have fun!