With your height of 1,82m and only 53 kg weight, your BMI is rather underweight. While this is not fit-for-all metric, it still can serve us a useful guideline for future improvements.
First of all you really need to sort out your diet. Since weight gain is only a function of caloric surplus  you need to eat and eat a lot more then what your are doing now. Granted, with such a slim body frame you might need less calories than average, but still 2500 you are eating is just enough for maintenance for an average guy. There are several protocols available for gaining weight and ideally you want to put on lots of muscle and as little fat as possible - a so called "clean bulk". But in your case i think it is rather acceptable to do a "dirty bulk" where you basically eat whatever you want, as long as you eat A LOT.
There is a lot of online calculators available where you can estimate your caloric maintenance level, i use this: http://users.telenet.be/WBtE/cunning.html
In average you'd add 10% to your caloric maintenance and see how it works but you can be a lot more aggressive with your parameters.
As far as supplements go, really all of your requirements (calories and nutrients) should be handled by natural foods. But considering that you are a vegetarian, supplementing creatin and protein is a feasible idea.
The importance of supplemental creatine is elevated in vegetarian and vegan diets due to the elimination of creatine’s main dietary sources 
Protein powders can be from either animal sources or plant sources, it is generally advised for vegans and vegetarians to consider protein supplementation due to the chance that their diet may be subpar 
The second is you training program. Since 4 months is an absolute beginner level, really any physical activity involving work (as exertion of force to move objects) will result in muscle tissue breakdown and repair, which is the way your muscles grow. But unless i'm reading your workout log incorrectly you are working one muscle group per day with 5 consecutive exercises, which is really just a waste of time, since by the second or third exercise your muscles are so fatigued that you can't use the weights heavy enough to cause muscle tissue breakdown. It's more akin to bodybuilders hi volume pre-contest routine for "pump" and muscle definition.
Ancillary exercises, which are by their nature inefficient isolation-type exercises, produce very slow progress. Anybody claiming rapid gains on triceps extensions or barbell curls is not utilizing particularly strict form and should be criticized for such foolishness 
Anyway, like it was noted before, heavy compound lifts are your best friends for both size and strength - squats, bench presses, deadlifts, pullups, rows, etc. I have nothing against StrongLifts, but Starting Strength is my favourite.
And since the best way to produce athletic improvement in novices is to increase strength, a program that increases total-body strength in a linear fashion is the best one for a novice athlete to use if he is to improve his performance the most in the shortest time possible 
Again, since you are an absolute beginner, first several months should produce the most rapid gains in size and strength in your lifting career, given proper nutrition and proper training program. If you are not seeing any results think of it as a good sign though too, at least you managed to realize that what you are doing now is incorrect and thus must be changed.
4 - M. Rippetoe "Starting Strength: Basic Barbell Training"
5 - M. Rippetoe "Starting Strength: Basic Barbell Training"
(PS: throughout the day i will try and add some links to proper sources and research material to support some of the claims, so it is not all broscience)