Back in the day before everyone wanted websites to be responsive, this was a lot easier. Setup some guides in PSD, slice the images, extract the colors and begin writing the html / css.
Now a days, I expect my designers to provide at least 2 versions of a site - 1 for mobile iOS and 1 full width desktop (generally around 1300px wide) - I'll figure out the break points in between for the array of android screen sizes and such.
My tips:
Practice, practice, practice! - it takes time to get good at slicing a PSD and turning it into html / css. Took me probably 5 or 6 sites to get really efficient at it and come up with a workflow. I've lost count now - have probably done 60-70 now. As anything else, it takes time to get good at.
Don't slice buttons, menu bars, backgrounds, etc... there are obviously exceptions to this, but any repeating colors in a menu bar, background, etc... get pure CSS. Buttons are generally CSS unless the designer did something really out there and I have no choice but to slice it. This really comes with experience and as you said, you struggled with a bit of CSS. I've been doing this for so long, I really don't have any tutorials for you to look at. Same for drop shadows - don't try to slice a drop shadow - if you can, use CSS.
Everything is a grid As you said, you've been playing with Bootstrap - everything uses a grid. Circles are masked areas in a grid, all divs can align to a grid, etc... Step back, look at the PSD, and place grid lines - start at the outside and work your way in. Small margin grid lines sometimes can't be avoided - this shows how good or bad the designer is. If elements are off by a few pixels, unless theres good reason for it, i'll push the element to the closest grid line (whatever looks best)
PSD slices help but sometimes, you just need to select, copy, new document, paste and trim it. Slices help in that you can name the slices, save for web and do save all slices. Throw out the garbage it generates and go from there.
Designers can be lazy, as programmers can be lazy I once has a PSD where everything was off by just 1 or 2px. It happens, but never take the designers word 100% and if you question them on it, they'll usually say "oh yeah, go ahead and move it over the 2px or whatever"
Fonts can be a pain We have web fonts now and most designers will use those - but learn what fonts are available as a web font, what looks good and if the designer uses something else, tell them it'll look better if they change it to this or that.
Stay away from apps or plugins that say "slice a PSD in 5 minutes flat!" they generate to much garbage and are a waste of time. Your goal is to take what you need from the PSD and write clean html / css. Not let some app that tries to cover everything do all that for you.
Eventually, if you do enough sites, you'll come up with some basic templates that'll help speed all of this up. Like I said, slicing a PSD is just another part of the process of building a site. Learning good html / css is apart of the game.