Does your gym have dumbbells? Because those are probably the best tool along with barbells.
Now, I've got the feeling you're NOT training for strength or performance but you're looking at generically getting in shape, put on some mass and such, correct me if I'm wrong.
If that's the case, I strongly suggest you look into using dumbbells as your primary tool (free weights) and then smith machine as a secondary tool (for muscle exhaustion). If you're worried about learning wrong patterns and motions, this sounds like the most sensible plan, so you can get the right patterns with dumbbells and then just reinforce muscles with the Smith.
Some examples of stuff you can do:
-chest: for aesthetics/mass, dumbbells are the way to go anyway. Presses variations at various angles of the bench, if your pseudo-gym has parallel bars, dips are more than a valid option. Also, look into pushups variations - some of the most effective are the plyo pushups and the hand release pushups (there's an article by CT about those here on T Nation). Honestly don't think Smith is necessary here, you can probably reap more benefits by doing plate raises variation and squeeze the plate as hard as you can during the reps, it tends to give chest a great pump.
-shoulders: shoulder press, push press and clean and press. Rear raises, plus possibly lateral raises (if you learn how to do them properly without butchering your shoulders). I suggest you buy an elastic band and do bands pull aparts and/or face pulls between sets of pushing exercises.
At the Smith: seated shoulder press.
-traps: one arm dumbbells snatch (that's a GREAT exercise all around), shrugs
At the smith: shrugs, but do them unilateral, one side at a time
-biceps: chinups have to be your base for biceps. Then the usual curls. Just focus on a very slow eccentric (lowering) portion of the lift and on keeping your body braced, don't use youre body to lift the weight on arms stuff.
Smith: nothing here
-triceps: dips are, again, your best bet if your gym has parallel bars. Various presses (shoulders and chest) will give you the necessary base. Triceps extensions or whatever they're called. Kickbacks are good too - just do them laying with your abs and chest against an inclined bench, and use both arms at the same time. This will prevent shoulders from rolling forward
Smith: nothing here too
-abs: hanging leg raises & variations, also take a look at Meadows' abs pulldown
-lats: Kroc rows, chinups/pullups
Smith: check Meadow's smith row
-legs (front): goblet squats, or you can clean two dumbbells from the ground to your shoulders and squat. Walking lunges with dumbbells, too
Smith: squats, front squats
-hamstrings/posterior chain: this is tricky, i don't think you can have a decent substitution to deadlifts (& variations) without a barbell, at least not directly.
What you CAN do, here, is to use the leg press machine and set your feet high on the platform so that the your toes are OUTSIDE of the platform, and keep your feet width a bit narrower than usual. Then, push with your HEELS. This stance will focus the movement on your hamstrings.
As for the posterior chain, enter the loaded carries. Farmer walks, suitcase walks, overhead carries, bag walks... do them. Depending on your schedule you might do one of them every workout, just rotate between the various types of carries. They'll help your posterior chain A LOT, keep your posture in check and strengthen your whole body while helping you burn fat, you just can't go wrong with them and really they're your best bet to have a decent posterior chain with your current equipment.
Smith: no clue
Now, how/when to fit the Smith Machine in all of this... simply do it AFTER free weight stuff. You won't learn "wrong patterns". If you goblet squat with proper technique (<<<relevant keywords here), you won't mess up your squatting pattern if you do front squats at the smith machine to isolate the muscles after you have done the goblet squat. Keep the weights on the Smith low and try to hit lots of reps. Like 5x10.
So basically just start off your workout with dumbbells and focus on keeping a good technique, slow negative, controlled pace. After you've done your exercises with dumbbells, move to the Smith and finish off the muscles. Easy as that.
You can either do free weights-Smith back to back (i.e. DBs shoulder press followed by seated Smith shoulder press), or you can do all your free weights stuff first and then move on to the Smith (i.e. on your pushing day you could do flat bench DB press, DB shoulder press, plyo pushups, rear raises, then move to Smith and do seated Smith shoulder press and unilateral shrugs).