I presume that you did pretty well in the course (say, B and above; otherwise it will be really difficult to continue) and you are looking for mathematics intense (mathematical sciences, computer sciences, physics, "mathematical" chemistry, "mathematical" economics etc) but not mathematical professional (mathematics specialist) education (mathematics specialist IMHO must take MAT267 rather than MA244 and APM351 rather than APM346).

So, what after MAT244? I am discussing only mathematical studies dependant on MAT244.

1) You can take PDE (Partial Differential Equations) course (studying in its framework some necessary topics of ODE (namely, boundary value problems for ODE)). Here we have APM346 and how it looked in Fall 2012 you can see on another "category" of this forum (and linked course webpages where complete lecture notes are provided). Currently timetable for Fall 2013 has not been released but I believe that I will continue to teach APM346.

2) You can continue ODE studying the rest of Chapter 9, Chapters 5, 6 and 11, may be Chapter 10 (elements of PDEs) of our textbook Boyce--DiPrima (an extended one) and also elements of Calculus of Variations. There is no class but one can take Reading Course (in this case I would write lecture notes for Calculus of Variations similar to those for APM346); some lecture notes for APM346 could supplement or replace Chapters 10, 11.

3) Both. Then there will be certain overlapping (Chapters 11 and 10). Chapter 11 of Boyce--DiPrima is crucial for PDE class (but it is basically covered in APM346), Calculus of Variations has PDE analogue (albeit not covered APM346) and thus is useful.

4) After APM346 reading course covering extra material is also possible.

PS. This is my personal opinion.