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.