I'm having trouble understanding when the -ig and -iĝ suffixes are needed and when they're not. For example, which of the following is correct (or are they both correct with different shades of meaning?):

La armeo detruis la urbon.
La armeo detruigis la urbon.

Same question regarding the following:

Dum la pluvego hieraŭ nokte, la arbo falis.
Dum la pluvego hieraŭ nokte, la arbo faliĝis.

I understand the meaning that these suffixes impart, but are there are any rules or guidelines about when to use them and when not to?

69UM24OSU12:I'm having trouble understanding when the -ig and -iĝ suffixes are needed and when they're not... are there are any rules or guidelines about when to use them and when not to?

The correct forms are:

La armeo detruis la urbon (detrui requires a direct object, namely la urbo).
Dum la pluvego hieraŭ nokte, la arbo falis (fali can stand alone).

There are, alas, no fixed rules that can tell you which verbs require an object ('transitive' verbs), and those that do not ('intransitive' verbs). However, a useful list of the commonly used ones is found on page 179 of Teach Yourself Esperanto, and for your convenience I list them here.

Transitive, requiring -iĝ to make them intransitive:
etendi, fendi, fermi, fini, kolekti, komenci, movi, renkonti, rompi, ŝanĝi, turni, veki, vendi.
Examples:
Mi vendis mian aŭton. I sold my car.
Tiu ero bone vendiĝas. That item sells well.

Intransitive, requiring -ig to make them transitive:
boli, bruli, ĉesi, daŭri, droni, halti, kreski, krevi, pasi, pendi, sidi, soni, sonori, stari.
Examples:
La bildo pendas sur la muro. The picture hangs on the wall.
Mi pendigis la bildon en la salono. I hung the picture in the lounge.

I recommend learning these two lists off by heart. It may well only take about half an hour of concentration, and then you won't have to worry about them (too much) again!

Miland:
There are, alas, no fixed rules that can tell you which verbs require an object ('transitive' verbs), and those that do not ('intransitive' verbs). However, a useful list of the commonly used ones is found on page 179 of Teach Yourself Esperanto, and for your convenience I list them here.

Thank you so much for your help. I found a copy of the book in a local library. (It's a shame how hard it is to find books on Esperanto for sale in the U.S. I have to shop mostly on line.) In English we switch verbs from intransitive to transitive without changing them- I walked home. / I walked the dog.- and this has been one of the more difficult concepts for me to grasp. The book helps to clarify it for me.

There is one thing that you have to be careful of when learning the -ig and -iĝ suffixes.

Some texts say that -ig is for making intransitive verbs transitive, and -iĝ is for making transitive verbs intransitive.

This is misleading, in the sense that, while it is true - and it certainly does work that way - that is not what they are FOR. The transitivity flip-flop is an effect, not a cause. The meaning of -iĝ is to indicate that the subject is what is being affected by the verb in the sense of "becoming" whatever it shows. The fact that that makes a transitive verb into an intransitive verb is simply a logical side-effect (since there is no direct object in such a statement, the verb acts intransitively by default).

The same goes for -ig: it shows that the subject caused the direct object to perform the action indicated by the verb (rather than the subject performing the action on the direct object). The fact that it changes an intransitive verb into a transitive one is, again, simply a side effect of the resulting construction. Because of the function of the suffix, it is possible to add -ig to an already transitive verb. The resulting verb is, of course, also transitive (but requires two direct objects instead of just one).

La letero skribiĝis.
The letter was written.
Skribiĝi = intransitive, no direct object.

Mi skribis la leteron.
I wrote the letter.
Skribi = transitive, one direct object.

Ŝi skribigis al mi la leteron.
She made me write the letter.
Skribigi = transitive, two direct objects.

In theory you could keep tacking -ig's onto a verb, each time adding one more layer of who is doing what to whom. However, you will probably never see such a construction in practice, because...

Li skribigigis al ŝi al mi la leteron.
He made her make me write the letter.
Skribigigi = Transitive, three direct objects, and quite incomprehensible. Take pity on your audience and word it differently.

-ig- and -iĝ- relate to change or transformation.

-ig- describes the process of causing change (to render)
-iĝ- describes the process of suffering change (to become)

As RiotNrrd explained, the changes in transitivity are a byproduct of that.

The real difficulty is that esperanto verbs can be either transitive or not by themselves (or at least they can be seen as such) but there is no way to know that without a dictionary; you have to remember by heart which of them are transitive and which are not.

If my understanding of the customary usage is correct, ig with a transitive verb normally means getting someone else to do something. Thus la maljuno pentrigis la domon probably means that he hired someone to do it.

However, as PMEG (see the note below the second box) indicates, when ig is attached to a transitive verb, the object of the resulting verb can be either the subject or the object of the corresponding simple verb. Thus Ŝi skribigis al mi la leteron is ambiguous; if the intended object of skribigi were the subject of the simple verb skribi, we would have 'She made me write the letter'. If, on the other hand, the intended object of skribigi were the object of the simple verb, we would have 'She had someone write the letter to me.'

For this reason it may be best to put the first of the two another way. It could be Ŝi igis al mi skribi la leteron.

But, if I were you (69UM24OSU12), I wouldn't worry too much about such technical issues at this stage.