Let's 1st understand what these planets represent on their own?

Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.

Moon - Moon represents Mind, Emotions, Mother, Local Government, Peace of Mind, Home Environment, Water, Milk etc.

Mercury - Mercury represents communication skills, marketing, calculative ability, younger siblings, hobbies, skills with hands, intelligence, quick decision making, logical thinking, youthfulness etc.

3 planetary conjunction - Whenever 3 or more planets are conjunct with each other in a house/sign then things related to that house/sign become a kind of life focus for person. It is because of huge accumulation of energy within 30 degrees of sky. It means not only these 3 planets' energy is going into this house/sign but the energy of at least 4 to 6 house/sign is going into this house/sign as these planets will be ruling some houses in chart. So, this house/sign with conjunction becomes almost the most important house in chart. During Career Consultation, I give a huge importance to any such house which has 3 or more planets in it.

House Ruler - Besides these karaka things, Sun, Moon & Mercury will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include things related to the houses at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas.

Meaning of Sun-Moon-Mercury Conjunction - Prima-facie, it shows a very communicative person as Soul & Mind both are getting impacted by Mercury which represents Communications. It also shows someone very much interested in Business or whose parents may be Entrepreneur which might have made him interested in Business, ex. a family business like scenario. They will be highly curious people and would love to collect lots of information. They may love travelling too and mind will be at peace while travelling. If Moon or Mercury or both are combust then we would have to take combust significance too. Basically, this is a person who is continuously wandering from this place to that place or this thought to that thought. They can have a strict or harsh way of speaking if Mercury and Sun are close degree-wise but an emotional way of speech if Mercury & Moon are close degree-wise. If Sun & Moon are close degree-wise then as we know it leads to strict parenting or troubles to Mother. Overall, this person can do well in Business and Communications. 

Who will win the Conjunction? - Moon easily get impacted by other planets. So, Moon here is impacted by Sun & Mercury. Also, Mercury, being a neutral planet, takes the character of planet it sits with. So, Mercury is also impacted by Sun. Hence, in the end, Sun is winning the conjunction as it will be impacting other 2 planets. 

Dignity & Strength - 

This conjunction can work best in Leo & Sagittarius as all planets are in good dignity here.

Although, Mercury is debilitated in Pisces, but I still take it as least concerning thing. Hence, Pisces can be a good sign too.

This conjunction can be most challenging in Scorpio as all planets are in weak dignity here. 

In other signs, at least 1 planet will be in good dignity, hence results of the conjunction will largely depend on which planet is in better dignity in a sign, planets mutual impact on each other, as discussed above, and the dashas.