It took me at least one year to achieve this, because meat is so prevalent in our society. A friend of mine was in India recently and he said he did not miss meat at all since the non-meat dishes were so great. I think a lot of it is just how good the quality of vegetarian dishes is.
I believe restaurant options are very important since people learn from those and mimic them at home. We need restaurants to offer more vegetarian and vegan dishes for people to see that they can have a rich (IMHO richer) diet without meat.
I don’t know how to nudge the restaurants though. Lower tax on organic/sustainable food can be one thing, or even better subventions for such restaurants and produce. But most importantly the meat and dairy industry needs to get under real supervision. Governments all around the world have been looking away for far too long.
Visit and ask for vegetarian or vegan dishes. Move on if there is nothing worthwhile; let the staff know if you've liked a particular dish or find the selection too limiting. Demand drives supply.
In the Netherlands the vegetarian option used to be a boring salad with goat cheese. Nowadays not having decent meat-free options means you can't compete with the rest. This is something only the cheaper restaurants aimed at lower socio-economic classes can afford to do (but only for now). The reason for this discrepancy is that this class of people (mostly blue collar workers and their families) tends to lag behind the rest of society a bit as habits shift towards more healthy alternatives (smoking is another example).
It was interesting to me leaving Europe and coming to China. In upper class restaurants here, they put meat in everything, even as a "seasoning" in veg dishes. I think it's seen as an indicator of wealth. At working class restaurants the cheapest dishes are all veg, or have only the barest scraping of ground pork or bone stock. I much prefer to eat out here than i did in Europe (and definitely the North America) where meat and dairy tends to be the default option for blue collar grub. I still don't really understand how that works, economically.