I started with 20 minutes of light jogging / intense walking.
3 sets of 15 on the leg press machine at 30kg.
3 sets of 15 on a step up block (raising both legs up)
4 sets of 10 lunges (approx - it was length of a corridor but guessing each length was ~10)
3 sets of 15 lat pull downs (can't remember the weight - probably 20 or 25kg)
1 and half sets of 10 assissted pull ups. I couldnt finish the second set only got to 5/6.
3 sets of 15 on the chest press at 15kg.
20 minutes on the bike machine.
Firstly, your warmup, while it is not bad it is a little long, 10 minutes is a good length for a cardio warmup, the extra 10 minute could be added to your cardio at the end. The reason is you do not want to be glycogen depleted in your muscle before you lift however when you deplete your glycogen stores while lifting you will force your body to access more fat reserves for energy during your cardio at the end.
For lifting the first set of each exercise you do should be a light warmup set to warm the specific muscle you are about to exercise, this set is in addition to your working sets. cardio warmup get you part of the way towards injury prevention but will not help prevent injury in specific muscles.
Leg Press
This exercise targets the Quadriceps (front of legs) but the target muscle can be changed by altering foot position on the plate, a high foot position will move some focus away from your quadriceps and onto your glutes (bum) and to a lesser extent hamstrings(back of legs). Unless you want to build size rather than strong tight muscles your reps are to high. As a beginner 10- 12 reps are a better option with a weight that you can only just finish the last rep on each set however lower reps (5-6) with an extra set would be even better as you gain experience and confidence.
Second Exercise
Sounds like you are doing calf raises, calves respond best to high reps so sets of 15 is good but if you feel you can do extra reps in each set then you need to hang onto some weights to make the exercise work for you.
Lunges
Lunges target the Quadriceps, the same muscle as your leg press, doubling up on the same target muscle is useful for bodybuilders wanting to hit a muscle in slightly different ways, it is not a great use of time for a beginners. Another option would be to do a hamstring specific exercise to balance the quadriceps training. If machines are your preferred option, single leg standing hamstring curls are a good option
http://www.exrx.net/WeightExercises/Hamstrings/LVStandingLegCurl.html however my favourite hamstring exercise is this
http://www.exrx.net/WeightExercises/Hamstrings/ASHamstringRaiseSelfFloor.html also known as a Nordic and comes with a warning that it does lead to extreme DOMS until you are used to doing them, they should be done as slow as possible for best results.
Lat Pull downs
Targets the Lats good basic exercise again, reduce the reps to 10-12 unless you are looking to build size, adjust weight up accordingly.
Assisted Pull Ups
These also target the lats which will already be tired from the pull downs, it would be better to swap this exercise with an exercise like seated rows, DB rows or inverted rows
which targets most of the other muscles in your back, engages the core, and works the back of the shoulders. The exercise can be made harder by elevating the feet or easier by bending the knees slightly. (The person in the video is my son, a dual Australian record holder)
Chest Press
A good machine alternative to bench press for chest, triceps and anterior delts (front of shoulders), as with the other exercises, drop the reps down and increase weight as needed.
Cardio at the end is good and could be a little longer if you shorten your cardio warmup, If time is limited at the gym HIIT would be a good option.
