A major problem with most early linear algebra classes is that they emphasize solving problems by hand using rref, as these methods do not translate well to computer calculations since they are unstable and thus can give inaccurate results. While engineering students may not understand the technical definition of stability, they should be taught that
- Stable algorithms are important for linear algebra on the computer
- Some simple algorithms (like rref) are good for human calculation, but not stable
With this in mind, when I teach a similar applied course, I plan my lectures around making sure I get as far as eigenvalue decomposition, QR factorization (if possible), and (perhaps most importantly) SVD and at least one good application of each. This allows students to see 2-3 good natural bases for calculations besides just the standard basis, and is enough background for most basic applied linear algebra. Getting as far as SVD is a doable challenge in a one semester course, so the necessary prerequisites determine pretty much everything else.
Strang's linear algebra textbook is one of the best known textbooks that follows this trajectory, but in my opinion it's a bit too advance (with not enough concrete calculations) for a first textbook for engineering students. (It would be an excellent resource for yourself as you're brushing up, as are Strang's own lectures.) Thus I agree with @MrProof and his suggestion of Lay's textbook for students, if you want to follow a textbook closely. It's concrete and has enough calculations to be more accessible for the population you are targeting. It's good to supplement any textbook with 3Blue1Brown's excellent series of videos on linear algebra, which give geometric intuition in a way that no static textbook page will be able to.