References on Linear Algebraic Groups/Lie Theory

I am more algebraically inclined, and so this will reflect in my suggestions.

1. Lie algebras appeared in the context of Lie groups, and as such, there were first defined over $\mathbb R$ and $\mathbb C$. Of course, we can define them over any field. Hence, you could investigate what happens when the field is not algebraically closed, or what happens when the characteristic is not zero.
2. Depending on your knowledge of representation theory, you can look at the classification of finite-dimensional representations of semisimple Lie algebras over $\mathbb C$. At some point, you encounter Verma modules, which in general are not finite-dimensional. This leads to the notion of the Category $\mathcal O$ of a semisimple Lie algebra. (On this topic, there is a book by James Humphreys.)
3. As you mentioned above, Algebraic groups are a natural place to look after you have studied Lie groups. However, your knowledge of classical algebraic geometry may be an obstacle to appreciating and understanding this topic. In the case you know close to nothing on this topic, I recommend Springer's book over the others.
4. Continuing on the last point, here is an idea of a possibly interesting topic for your essay: there are strong connections between Lie groups and Algebraic groups in the structure and the ideas. What can we say about algebraic groups when we work over an algebraically closed field of characteristic zero? (This is Chapter 5 in Humphreys' book.)

Added: In my opinion, there are two ways to efficiently learn about algebraic groups:

1. Probably the most obvious: take a course on the topic. However, I know that this can be complicated (for example, in your situation, you have to learn about it more quickly, but also, such a course is not offered everywhere). However, the lecturer will be able to give insight into the theory and possibly applications, which are very valuable.

2. Once you know the basic terminology (say the first four chapters of Humphreys'), pick a random chapter in the book and start reading. Or read papers/books where algebraic groups are being used and see how the structure theory is used in concrete applications (personally, this was achieved through reading papers on $p$-adic groups, and later, when learning about automorphic forms and groups). Since you have taken a course on Lie groups, the structure theory should not surprise you (except maybe the fact that the focus is shifted from semisimple groups to reductive groups). The difficult part is proving the theorems we want, and this requires a good knowledge of algebraic geometry.

What about the combinatorics of Weyl groups? I don't know how much that is covered in your course, but this is a rich topic with direct connections to Lie Theory. Bjorner and Brenti's Combinatorics of Coxeter Groups and Humphreys' Reflection Groups and Coxeter Groups are two very readable texts which give a fairly comprehensive introduction to the subject. The other standard reference is Bourbaki's Lie Groups and Lie Algebras: Chapters 4-6, though this is a bit more daunting to someone first learning the subject.

This may be a bit late for the discussion, but what about: A.L. Onishchik and E.B. Vinberg (1990), Lie Groups and Algebraic Groups?