For what it's worth:
I just memorize one Pythagorean identity and one of the sum identities. Many of the others (besides the obvious ones: the reciprocal, periodicity, and Pythagorean) can be derived
starting with one of the sum formulas.
So, you could just memorize how to derive them. Of course, in a test scenario, this may waste precious time...
Reciprocal identities
The reciprocal identities follow from the definitions of the trigonometric functions.
$$\eqalign
{ \sec\theta&= {1\over \cos\theta} \qquad \tan\theta= {\sin\theta\over \cos\theta} \cr
\csc\theta&= {1\over \sin\theta} \qquad \cot\theta= {1\over \tan\theta} \cr
}
$$
Periodicity relations
The Periodicity relations follow easily by considering the
involved angles on the unit circle.
$$\def\ts{}\eqalign
{ \sin(\theta)&= \sin(\theta \pm2k\pi) \qquad \csc(\theta)= \csc(\theta \pm2k\pi) \cr
\cos(\theta)&= \cos(\theta \pm2k\pi) \qquad \sec(\theta)= \sec(\theta \pm2k\pi)\cr
\tan(\theta)&= \tan(\theta \pm k\pi)\phantom{2} \qquad \cot(\theta)= \cot(\theta \pm k \pi) \cr
}
$$
$$\eqalign
{ \sin(\theta)&= - \sin(\theta -\pi) \qquad \csc(\theta)= - \csc(\theta -\pi) \cr
\cos(\theta)&= - \cos(\theta -\pi) \qquad \sec(\theta)= - \sec(\theta -\pi) \cr
\tan(\theta)&= - \tan(\theta -\ts{\pi\over2}) \qquad \kern-3pt \cot(\theta)= - \cot(\theta -\ts{\pi\over2}) \cr
}
$$
Pythagorean Identities
The first Pythagorean Identity follows from the Pythagorean Theorem (look at the unit circle). The other two
Pythagorean Identities follow from the first by dividing both sides by the appropriate expression (divide through by $\sin$ or by $\cos$ to obtain the other two).
$$\eqalign
{ \sin^2\theta +\cos^2\theta&=1\cr
1+ \cot^2\theta& =\csc^2\theta\cr
\tan^2\theta + 1& = \sec^2\theta}
$$
Sum and difference formulas
Memorize the first sum and difference formula. The second one can be derived from the first using the fact that $\sin$ is an odd function.
One can then derive the last two sum identities by using the first two and the fact that $\cos(\theta-\pi/2)=\sin\theta$.
$$\eqalign{
\cos(x+y)&=\cos x\cos y-\sin x\sin y\cr
\cos(x-y)&=\cos x\cos y+\sin x\sin y\cr
\sin(x+y)&=\sin x\cos y+\sin y\cos x\cr
\sin(x-y)&=\sin x\cos y-\sin y\cos x\cr
}
$$
Double angle formulas
The Double Angle formulas for $\sin$ and $\cos$ are derived by using the Sum and Difference formulas by writing, for example $\cos(2\theta)=\cos(\theta+\theta)$ and using the Pythagorean Identities for the $\cos$ formula (I suppose the formula for $\tan$ should be memorized).
$$\eqalign{
\sin(2\theta)&=2\sin\theta\cos\theta \cr
\tan(2\theta)&= {2\tan \theta\over 1-\tan^2\theta } \cr
\cos(2\theta)&= \cos^2\theta-\sin^2\theta \cr
&=2\cos^2\theta -1\cr
&=1-2\sin^2\theta\cr
}
$$
Half angle formulas
The Half-Angle formulas for $\sin$ and $\cos$ are then obtained from the Double Angle formula for $\cos$ by writing, for example, $\cos\theta=\cos(2\cdot{\theta\over2})$
The $\tan$ formula here can easily be obtained from the other two.
(Note the forms for the $\cos$ and $\sin$ formulas. These aren't to hard to memorize)
$$\eqalign{
\cos{\theta\over2}&= \pm\sqrt{1+\cos\theta\over2}\cr
\sin{\theta\over2}&= \pm\sqrt{1-\cos\theta\over2}\cr
\tan{\theta\over2}&=\pm\sqrt{1-\cos\theta\over1+\cos\theta}
}$$
Well, since parentheses exist precisely to specify the intended order of operations in case the usual default rules don't cut it, it makes sense that they come first
As for exponentation, I'd say that this is a consequence of using superscripts to indicate exponentation, since those (via font size) provide a natural grouping. It'd certainly be very weird if $a^b + c$ meant $a^{(b+c)}$ instead of $(a^b) + c$, since the different font sizes of $b$ and $c$ indicate that they're somehow on different levels.
As MJD pointed out though, this arguments only applies to the exponent. Font size alone doesn't explain why $a + b^c$ means $a + (b^c)$ and not $(a + b)^c$ and the same for $a\cdot b^c$ vs. $a\cdot(b^c)$ respectively $(a\cdot b)^c$. For these, I'd argue that it's also a matter of visual grouping. In both $a\cdot b^c$ and $a + b^c$, the exponent is written extremely close to the $b$, without a symbol which'd separate the two. On the other hand $a$ and $b$ are separated by either a $+$ or a $\cdot$. Now, for multiplication the dot may be omitted, but it doesn't have to be omitted, i.e. since $ab$ and $a\cdot b$ are equivalent, one naturally wants $ab^c$ and $a\cdot b^c$ to be equivalent too.
For multiplication, division, addition subtraction, I always felt that the choice is somewhat arbitrary. Having said that, one reason that does speak in favour of having multiplication take precedence over addition is that one is allowed to leave out the dot and simply write $ab$ instead of $a\cdot b$. Since this isn't allowed for addition, in a lot of cases the terms which are multiplied will be closer together than those which are added, so most people will probably recognize them as "belonging together".
You may then ask "how come we're allowed to leave out the dot, but not the plus sign". This, I believe is a leftover from times when equations where stated in natural language. In most langues, you say something like "three apples" to indicate, well, three apples. In other words, you simply prefix a thing by a number to indicate multiple instances of that thing. This property of natural languages is mimicked in equations by allowing one to write $3x$ with the understanding that it means "3 of whatever $x$ is".
Best Answer
PEMDAS is an acronym to help you remember. try different forms of mnemonic devices, like acrostics: Please Excuse My Dear Aunt Sally; Pancake Explosion Many Deaths Are Suspected; Purple Egglants Make Dinner Alot Sickening; Pink Elephants March, Dance, And Sing; Pizza ended my donuts addiction Saturday