The first and last thing artists need to keep in mind about applying for a grant is that they may not get it. Rejection leads some artists to bitterness and dejection, an I-won't-get-burned-twice attitude that cuts off future options and attempts. The failure of a grant application to be approved may be taken personally as a negative judgment about the artist and his or her work or talent. Meanings are read into form letters and into the psyches of those who approve or disapprove applications -- many of whom serve as jurors on selection panels for only one grant cycle -- are examined to discern prejudices and favoritism. Hell hath no fury like an artist scorned.
It is wiser to realize that all funding sources these days are besieged by applicants -- that's the reason for the form rejection letters -- and that grantgivers need to look for reasons not to allocate money. Applicants who don't fill out the grant form completely or neatly, who are not residents of the right area (for instance, a German applying for a National Endowment for the Arts fellowship or a Maine resident seeking project funding from the Oregon Arts Commission), or who are not professional artists but art students, who are applying for money from the wrong organization (such as seeking a fellowship from the New York State Council for the Arts, which makes no individual grants), whose projects are not described succinctly or whose budget sounds inflated (or too low) -- these are the easiest to dismiss and are eliminated eagerly as the decision-makers look to whittle down the stack.
Artists should seek information about the granting process before they apply and after the decisions are made. Public agencies especially are obligated to provide information on why a particular proposal was rejected (written notes are taken and filed), and that feedback is welcome. Problems in an application or an overstated budget or a lack of documentation or something else may have been the reason, and they can be corrected for the next round of funding or when applying to another granting agency.
...which is why I encourage entrepreneurialism over fundraising
When I entered the arts service field, in the mid-1970s, the most commonly asked question was how and where to apply for grants, and most of the career books for artists of that time and into the 1980s devoted the majority of their pages to this topic. Perhaps the reason was that public arts agencies were being created on the national and state levels during the 1960s and '70s, and their budgets increased every year. Career seminars for artists during this time also focused almost exclusively on the grant application process and, still today, the most common form of "technical assistance" that state arts agencies provide to artists and arts organizations is information on how to fill out their grant application forms. (In fact, state arts agency websites primarily exist to allow applicants to download applications: The Internet is a labor-saving tool, rather than an interactive one, for these agencies.) These days, the question of how and where to apply for a grant seems less central to the careers of most serious artists. Ongoing and increasing public financing of the arts appears to be less assured than it did just 20 or 30 years ago. The budgets of many public arts agencies and some private foundations have been reduced or just not kept up with even modest inflation, and the competition for every available dollar is ferocious. Artists who tie their career hopes to grants and awards are likely to meet with repeated frustration. Instead, artists must see themselves as entrepreneurs, learning how to earn rather than simply apply for money.