Also, are Japanese to Japanese dictionaries better than English to Japanese?
There isn't really a specific time when you suddenly become able to use monolingual dictionaries. But I can make a few points I think are relevant:
If you can't read many kanji, you're going to trouble using most monolingual dictionaries. If this is the case, you might consider starting with a freely available online dictionary like 大辞林 or 大辞泉 and using electronic tools to help you look up kanji faster.
There are also a few monolingual dictionaries targeted at Japanese children, such as the チャレンジ小学国語辞典. These dictionaries use simple words and furigana over all the kanji. This is another option if you're a beginner who can't read many kanji yet.
Monolingual dictionaries are almost exclusively designed for native speakers. This is very unlike the situation in English, where there are many good monolingual "learner's dictionaries" on the market (Macmillan, Longman, etc.) designed to meet the needs of non-native speakers.
This is also true of most of the good bilingual dictionaries, which are designed for native speakers of Japanese learning English. Unfortunately, most of the bilingual dictionaries targeted toward learners of Japanese aren't very good, although there are a few exceptions (like Kodansha's Basic English-Japanese Dictionary, which I think is good for beginners).
You can use both monolingual and bilingual dictionaries. Monolingual dictionaries tend to define things more usefully, although that depends on the dictionary (I think 明鏡国語辞典 is one of the best, and it's available in electronic versions).
Very large bilingual dictionaries tend to have lots of example sentences in Japanese and English side-by-side, and even though their definitions tend to be less helpful than those in monolingual dictionaries, the examples can be quite a bit more helpful. After all, we learn by example; it's not enough to simply read a definition, no matter how good that definition is.
I suggest trying out monolingual dictionaries as soon as possible. If they seem too hard for now, you can always come back to them in a few months, or you can try to find an easier dictionary (like the チャレンジ dictionary for kids). Don't worry! It'll get easier as you practice :-)