There are two distinct questions here. As pointed out by gauss, the "equicharacteristic" case (when you look at Sylow $p$-subgroups for the defining characteristic $p$) is fairly straightforward, since it's easy to compute the index of a Borel subgroup in a finite group of Lie type. Standard structure theory (in terms of BN-pairs) found in many sources shows that a Borel subgroup is the Sylow normalizer in this case.
For other primes $r$ dividing the group order, it's much harder to make general statements about the number or structure of Sylow $r$-subgroups. Here it's very useful to study a comprehensive summary of properties of the known finite simple groups: Number 3 (1998) in the series of AMS monographs by Gorenstein, Lyons, Solomon The Classification of the Finite Simple Groups. See in particular sections 3.3 and 4.10 for the two types of primes. This volume has lots of other information about the groups of Lie type, including their orders.