Arguably à is the most important preposition in French: it has at least six distinct meanings or functions. It can denote location (in, at), destination (to), a time (at x o’clock), distance (à 10km de, 10km from), possession (un ami à moi), measurement (acheter au poids/au kilo – by weight/the kilo), manner, purpose, style etc. (aller à pied – go on/by foot, sac à dos – rucksack, la vie à la française – the French way of life).

à, en, dans denoting location/destination – place-names
À is nearly always used with names of towns (je vais/vis à Paris, au Havre). En Avignon and en Arles are common provincialisms in Provence which should be avoided (they often sound affected). Dans can however be used to demarcate an area in which something happens, eg, il se rend à Paris en voiture mais il circule dans Paris à vélo (he drives to Paris but he gets around Paris by bike).

à, chez, en, dans denoting location/destination – places
À is commonly used for places (aller/être à l’école, l’usine, au bureau), but some nuances are worth noting:
• il est à son bureau (implying the workplace, the building); il est dans son bureau (implying the room he uses as his office); il est au bureau (he’s at the office, at work, ie, not at home).
• Chez is used with names of people/professions. Compare Je vais au salon de coiffure with Je vais chez le coiffeur (chez le médecin, dentiste).
• Il travaille à domicile (he works from home), but Il travaille dans le secteur sanitaire (in the health sector).
• Dans tends to be more precise than en; the former is used with an article, possessive etc. Compare:
Les enfants sont dans la classe (…in the classroom, ie, not elsewhere) but Les enfants sont en classe (…at school);
un incendie s’est déclaré dans la prison (a fire broke out in/at the prison) but Il est en prison (he’s in prison)

à, en with means of transport
The distinction is fairly clear in French. If you can get ‘on’ the means of transport, use à, but if you get/travel in it (even though we say ‘get on’ a train/bus in English) then use en/dans.
Se promener à cheval, à bicyclette, moto (ride a horse/bike…), although en vélo/moto is also widely accepted; but note that ‘get on’ here is monter sur (un cheval/vélo – mount a horse/get on a bike)
Voyager en train/avion/voiture/bateau (travel by…), monter dans l’avion/le train/bus/voiture (to board the plane, get on the train/bus, get into the car).

à denoting possession
This is mostly to be avoided when followed by a noun:
la maison de mon ami, never à – considered slovenly except in certain idioms (un fils à papa – a daddy’s boy). It is mandatory when followed by a personal pronoun:
un ami à moi – a friend of mine)
and when the verb être is used to denote possession:
Ce chien est à Richard (that’s Richard’s dog).
Also, note the subtle difference between à and de, as in:
Ce livre est de lui (this is his book = he wrote it) cf
Ce livre est à lui (this is his book = he’s bought – or has copyright on – it).