When I first see an integral, substitution is a technique that comes to mind first. I'm sure you've learned "u-sub", as some textbooks like to claim that the two fundamental techniques are "u-sub" and integration by parts.
Personally, I prefer substitution because it is usually very flexible and elegant. Some textbooks will try to make "u-sub" and trig sub seem like very different techniques (sometimes giving formulas for trig sub) but they are essentially the same technique based on the chain rule. Typically if you have a square root in the denominator and that's the ONLY expression in the denominator, you want to try a trig substitution (it's easy to find the correct substitution, since it is based off of the pythagorean identities). However, that is not always the case, and it's not always a good idea to generalize trig sub to "square roots". If there are two expressions multiplied in the denominator, one with square roots, u-sub may work better. Sometimes you won't find square roots but a trig substitution would work nicely (think of the pythagorean identities).
Improper integrals are pretty easy to spot. Look for infinity in the limits of integration and keep in mind where the denominator of the integrand goes to zero.
Denominators that factor nicely lend themselves to partial fractions. There is also a quick way of determining the coefficients of the separate expressions. As you may have noticed, paying attention to the denominator of any integrand is important.
Integration by parts (IBP) is somewhat of a last resort for me. It's pretty easy to see if all other methods will fail or be inefficient. IBP is based on a simple derivation, but in many cases it's a pretty dull and inelegant method. Anyways, to determine u and dv, it helps to think about which functions are difficult to integrate or easy to differentiate and vice versa. Consider natural logarithm functions. It's hard to find the antiderivative, but very very easy to differentiate. If you see a natural logarithm, it's usually a good idea to make that your "u" (make sure you see why). Inverse trig functions are also rather annoying to integrate, but they are easy to differentiate, so that's also a good choice for "u".
Polynomials are in the middle of the spectrum. They are easy to integrate and differentiate. Use this to you advantage (u or dv, depending on what the other function is). Exponentials are very easy to differentiate, and fairly easy to integrate. Considering that a natural log or inverse trig expression may be the other function, you'll want to select the exponential function as "dv" (between a polynomial and an exponential, use your judgement). Lastly, certain trig functions are very easy to integrate, and easy to differentiate. As in the case of the exponential, you'll usually want to make this "dv" if paired with a natural log or inverse trig function. When it's a trig function and a polynomial, letting "u" be the polynomial has the advantage that you'll probably end up having to integrate an expression of similar or usually simpler terms (because the du in v*du will be of one less power than u). use your judgment for a trig function and an exponential.