Custom Date Package

Here's the code for mycustomdate.sty:

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{mycustomdate}[2014/03/19 1.0 My custom date format]

\RequirePackage{etoolbox}%
\RequirePackage{pgfkeys,pgfcalendar}

% Define an ordinal command:
\newcommand*{\ord}[1]{%
  \number#1%
  \ifcase#1\or st\or nd\or rd\or th\or th\or th\or th\or th\or
  th\or th\or th\or th\or th\or th\or th\or th\or th\or th\or
  th\or th\or st\or nd\or rd\or th\or th\or th\or th\or th\or
  th\or th\or st\fi
}

% Define registers needed by \printdate:
\newcount\julianday
\newcount\dayofweek

% Define generic date format:
\newcommand*{\datefmt}[4][]{%
  \ifstrempty{#1}
  {}% day of week missing
  {%
    \pgfcalendarweekdayname{#1}\space
  }%
  \ord{#4}~\pgfcalendarmonthname{#3} \number#2%
}

% Define command to convert ISO date to a form that can be used by \datefmt:
\newcommand*{\printdate}[1]{%
  \pgfcalendardatetojulian{#1}{\julianday}%
  \pgfcalendarjuliantodate{\julianday}{\thisyear}{\thismonth}{\thisday}%
  \pgfcalendarjuliantoweekday{\julianday}{\dayofweek}%
  % Now display the date:
  \datefmt[\dayofweek]{\thisyear}{\thismonth}{\thisday}%
}

% Redefine \today to use the same format:
\renewcommand*{\today}{%
  \pgfcalendardatetojulian{\year-\month-\day}{\julianday}%
  \pgfcalendarjuliantoweekday{\julianday}{\dayofweek}%
  \datefmt[\dayofweek]{\year}{\month}{\day}%
}

% Check if babel is used with the british option:
\AtBeginDocument{%
  \ifdef{\datebritish}% check if \datebritish exists
  {% it does exist, so redefine it
    \renewcommand*{\datebritish}{%
      \renewcommand*{\today}{%
        \pgfcalendardatetojulian{\year-\month-\day}{\julianday}%
        \pgfcalendarjuliantoweekday{\julianday}{\dayofweek}%
        \datefmt[\dayofweek]{\year}{\month}{\day}%
      }%
    }%
  }%
  {}% doesn't exist, do nothing
}

\endinput

And here's the code for the sample document:

\documentclass{article}

\usepackage{mycustomdate}

\begin{document}

Today: \today.
Tomorrow: \printdate{\year-\month-\day+1}.
Yesterday: \printdate{\year-\month-\day+-1}.

The first day of this month: \printdate{\year-\month-1}.
The last day of this month: \printdate{\year-\month-last}.

A specific date: \printdate{2014-3-20}.

A date without the day of week: \datefmt{2014}{3}{20}.

\end{document}

Download customprintdate.tex or customprintdate.pdf.