I have yet to teach such a course, but I would motivate abstract algebra based on plane geometry, especially group theory. It certainly worked for me to some extent, though I suspect that's because I saw a lot more plane geometry in high school than our students ever do. Various flavors of plane transformations, dihedral groups, regular polyhedra and crystalline structures, you can even go into combinatorics with Polya enumeration$\dots $ I think a geometric picture would be valuable even after moving away from geometric problems.

As for analysis, I would go for the big theorems, which seems where the difference between analysis and calculus lies. For instance, after doing a bunch of limits of integrals depending on an integer parameter, wouldn't it be nice to know that limits and integrals can be interchanged when the convergence is uniform? The big theorems can be presented not for the sake of abstraction, but because they make boring computational tasks easier.

I don't know if this answers your question, though I'm putting it out there anyway; this is almost too obvious, you probably wanted more details or a different tack altogether.