A note before we begin: Anatolia, Syria, and Palestine are where most of the fighting occurred during the First Crusade. These regions, while arid compared to western Europe, are not deserts, and were probably even less so at the time, given that the earth as a whole has warmed since the 11th century. Anatolia is heavily mountainous, with fertile valleys between the mountains. The Levant (that is, Palestine and Syria) has a Mediterranean climate, especially near the coast, and was a fertile, agriculturally productive, heavily populated region.

As to the real meat of your question? I'm going to quibble a bit and tell you that our sources, literary or artistic, aren't great. We know much more about what the elite of the army would be wearing. Presumably equipment would exist on a continuum between the very well-equipped knights, and the minimally equipped foot soldiers, but it's hard to define an average.

Basic attire, regardless of social class, would consist of one or more roughly knee-length tunics (also called a kyrtle) of linen or wool, linen braies (baggy boxer shorts, essentially), wool chausses (leggings) tied to the braies, soft leather turnshoes, and a wool cloak as outerwear. The amount of clothing worn would, of course, depend on availability and weather; more layers when it was cold and fewer when it was hot. There are medieval images of peasants working in the fields in nothing but their chausses and braies.

Weapons would vary between soldiers. While a knight might well go off on crusade with lance, sword, dagger, and even an axe or a mace, the poorest soldiers in the army might only possess a spear or a bow. The crossbow was generally the weapon of a professional, relatively high status soldier.

The best equipped men in the army - whether knights or wealthier footmen - would have worn a knee-length mail hauberk with short or long sleeves and an integrated hood called a coif. The helmet would be an open-faced nasal helm of either dished or multi-piece (spangen) construction. Padded garments are not directly attested to in the First Crusade, but we know crusaders were wearing them by the end of the 12th century, either in addition to or in place of mail. Large kite shields would be a near-universal feature.