'Ello Snowflake
The whole matter can be very complicated to explain fully, but the usual gist can be found in the following explanation (approximately!)
It might not be absolute, but it does give a general idea. I'll try to put it as simply as I can ^^' :
(1)
In some cases when you combine two or more kanji to make another word, the 'dots' are added to the suffix.
手紙 (てがみ teGAmi - letter) --- 手 (て te - hand) and 紙 (かみ KAmi - paper)
立花 (たちばな tachiBAna - [a surname]) --- 立 (たち tachi) and 花 (はな HAna)
Similarly, ぢ (dji/di) is normally seen in words that were originally pronounced as ち (chi/ti).
Examples include:
鼻血 (はなぢ hanaDJI - nosebleed) --- 鼻 (はな hana - nose) and 血 (ち CHI - blood)
身近 (みぢか miDJIka - familiar [to oneself]) --- 身 (み mi - body) and 近 (ちか CHIka - close)
Similarly for づ (dzu/du) and つ (tsu/tu):
手作り (てづくり teDZUkuri - handmade) --- 手 (て te - hand) and 作り (つくり TSUkuri - to make)
三日月 (みかづき mikaDZUki - crescent moon) --- 三 (み mi - three), 日 (か ka - day) and 月 (つき TSUki - moon)
Also:
詰め合い uses つ tsumeai; 理詰め uses づ ridzume
付け上がる uses つ tsukeagaru; 気付ける uses づ kidzukeru
地下 uses ち chika; 有漏地 uses ぢ urodji
As far as I know, づ dzu and ぢ dji are definitely used as suffixes.
(2)
Syllable preceded by its 'dotless' form:
縮む (ちぢむ chidjimu - to shrink) --- not ちじむ (chijimu)!
鈴 (すず suzu - bell) --- not すづ (sudzu)!
続き (つづき tsudzuki - continuation) --- not つずき (tsuzuki)!
(3)
Stand alone words in the ji/dji case are, to my knowledge, generally written as ji じ.
The following are all pronounced ji, with a kana form ji じ:
字、地、痔、辞 etc.
As for zu/dzu, they are generally zu ず:
図 (I can only think of this one as far as common words, taking into account of usage, are concerned)
(4)
Finally, words that have a different reading on its own than being in a compound word would usually take じ and ず instead of ぢ and づ:
寺 (てら tera - temple) / 龍門寺 (りゅうもんじ ryuumonji - [a temple name])
時 (とき toki - moment) / 時間 (じかん jikan - time)
豆 (まめ mame - bean) / 伊豆 (いず izu - [a place name])
頭 (あたま atama - head) / 頭巾 (ずきん zukin - hood)
In general, it will come with experience which one to use.
It's not difficult actually, it's one of those things you'll remember for life once you learn it :3
Sorry for so much information, but the whole topic is very wide. Even native speakers would mix up the uncommon words.
If I have not explained it clearly enough please feel free to ask!