The theory of intrinsic Diophantine approximation concerns the problem of approximating points on a variety by rational points lying on the same variety. We first consider this problem from a metric point of view, where we try to derive results regarding almost all points in the sense of measure. In this direction, we derive a zero-infinity law for Hausdorff measure on certain varieties using a projection argument. The key novelty here is the use of algebraic geometry to control the complexity of the rational points under the projection. Following this, we turn to the problem of finding algorithms for Diophantine approximation on varieties. This is inspired by a problem of constructing efficient universal quantum computers. We consider continued fractions in terms of a certain tree, and use this description to describe the problems in constructing continued fractions for the unit circle. Finally, we consider a problem of finding transcendental numbers which behave like Pisot numbers. We use finite automata to construct transendetal numbers with known Diophantine properties and do a computer search through these.