An introduction to Khovanov homology, Heegaard-Floer homology
I highly recommend that you begin reading about Khovanov knot homology from the works of Dror Bar Natan.
In particular,
- On Khovanov's Categorification of the Jones Polynomial, Algebraic and Geometric Topology 2-16 (2002) 337-370.
- Khovanov's Homology for Tangles and Cobordisms, Geometry and Topology 9 (2005) 1443-1499.
His exposition is clean, intuitive, and motivated by the geometric/cobordism perspective. I dare say it's fun to read.
Regarding, Heegaard-Floer Knot Homology, I'd go to the source: Ozsváth and Szabó.
A very good, more recent introduction to Khovanov homologies was written by Dolotin and Morozov.
- Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial.
- Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials.
- Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants.
Regarding Heegaard Floer: For the "original" theory, I would recommend "Heegaard diagrams and Holomorphic disks" first (not necessarily the whole article...), then "An introduction to Heegaard Floer homology" (both by Ozsvath and Szabo). At least the latter might be a bit difficult as they leave some nontrivial results unexplained. Also perhaps "Combinatorial Heegaard Floer homology and nice diagrams".
There's also a combinatorial HF theory based on a construction from what's called Grid Diagrams. The advantage is that you don't need much of a geometric background (holomorphic disks etc.). On the other hand, you need some advanced algebraic knowledge, and it's also very different from the geometric approach in the articles above, i.e it won't really help you understanding it. For this approach, try "On combinatorial link Floer homology" by Manolescu.