Luckily, I am studying this forgotten branch of mathematics. The following references are must-read according to my knowledge:
Irving Kaplansky, Introduction to differential algebra,
Hyman Bass and Alexandru Buium et.al, Selected Works of
Ellis Kolchin with Commentary, American Mathematical Society, 1999
Buium, Differential Algebra and Diophantine Geometry,
These three references are ordered according to their difficulty. The first book is the only book which can be read with bare knowledge of Lie theory. Since you mentioned you are familiar with algebraic geometry, the second book is a good guide towards the papers written by Kolchin which are obscure due to the evolution of notions and concepts throughout these years. And the third book points out the recent trends in this branch and requires more prerequisite and mathematical maturity. If you are not sure whether you want to put so much effort into this branch this will be a good read: A first look at differential algebra
Ellis R. Kolchin, Existence theorems connected with the
Picard-Vessiot theory of homogeneous linear ordinary differential
equations, Bull. Amer. Math. Soc. 54 (1948), 927–932
References E. R. Kolchin, Galois Theory of Differential Fields ,
American Journal of Mathematics, Vol. 75, No. 4 (Oct., 1953), pp.
References E.R.Kolchin, Algebraic Matric Groups and the
Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential
Equations , Annals of Mathematics, Second Series, Vol. 49, No.1
(Jan., 1948), pp.1-42
Hopefully my answer will help, and my question is here.
Why differential Galois theory is not widely used?
For more advanced discussions on this topic, there is a MO post Why differential Galois theory is not widely used?(MO version).
An interesting related MO post is Does a theory of stochastic differential algebras exist?. In this post I briefly explained difference between differential algebra and stochastic calculus.