Aquinas's first two ways may be seen as versions of the cosmological argument. The basic idea is that everything has a prior cause, but the chain of causes can't go back infinitely far, so there must be a first cause. The "first way" argument might be summarized like this:
1. Some things change. (empirical premise, verified by observation)
2. Everything that changes is made to change by something else. (Aquinas has a separate argument for this)
3. The chain of causes can't go back to infinity.
4. Therefore, there must be a cause of change that does not itself change.
Premise 2, that everything that changes is made to change by something else, can be seen as having two parts: that every change is caused, and that the cause of a change in an object must be a different object from the one that changes. Aquinas seems to presuppose the first of these two claims without argument. The argument for the second appears to be this:
1. An object can change from not having property G to having G only if the object is potentially G but not actually G.
2. The cause of an object's becoming G must itself actually be G.
3. Therefore, a thing cannot cause itself to acquire a property.
There are several points at which these arguments might be questioned. Consider premise 2 of the first argument. The second argument is supposed to show that 2 is true. However, the second argument shows at best only that if a change has a cause, the cause must be a different object from the one that changes. It doesn't show that there must be a cause of change at all. And in fact the best evidence we have from contemporary science suggests that in fact changes do not necessarily have causes. Consider the decay of an atom of a radioactive substance. It appears that there simply is no explanation for why this happens at one time rather than another: when such an atom will decay is a purely random matter.
Another weak point is premise 2 of the second argument. Aquinas's example is heat: one object can become hot, he suggests, only if a second object which is already hot causes it to become hot. But this isn't the only way something can become hot! Think of chemical reactions that produce heat, or of producing heat by means of friction.
It may also be worth noticing that if the argument were successful, it would show the existence of at least one unchanged changer -- but nothing in the argument shows that there aren't more than one.
Aquinas's "second way" is closely related to the first; it's a more general version of the cosmological argument. The argument is something like this:
1. Every sensible event has a distinct cause.
2. Either the chain of causes goes on forever, or there is a first cause.
3. The chain of causes can't go back forever.
4. Therefore, there is a first cause.
This has some of the same problems as the previous version -- in particular, it is not at all clear that every sensible event has a cause (the example of radioactive decay applies here also).
Apart from questions about whether the arguments are sound (i.e. whether their premises are true and their conclusions follow from their premises), there is also the question of whether the conclusion really proves what the theist wants. The conclusion is that there is a first (uncaused) cause. But why should we identify this with God? As Russell points out, if the argument works then there must be something that is uncaused, so the first premise is false. But once we've accepted the possibility that something is uncaused, it seems that we've opened the door to the idea that something other than God is uncaused (for example, the Big Bang).
Aquinas's "Fifth Way" is a version of the "teleological argument" or "argument from design." This argument is, I think, best thought of as an "inference to the best explanation" rather than as a deductive argument (although Aquinas seems to think of it as deductive). The argument appeals to the existence of order and apparent purpose in the universe, and argues that the best explanation of this order is that the natural world was designed by an intelligent being.
Until a hundred and fifty years ago, more or less, intelligent design was the only explanation available for much of the apparent purpose in the natural world. Since Darwin, we have a serious competing explanation, however. The idea here is that apparent purposiveness in nature is to be explained as the end result of very long periods of random mutation and natural selection.