Travelling to a new country, let alone moving to one, has to be among the most daunting of tasks for non-meat eaters. Aside from the language barriers that often make communicating particular culinary tastes difficult, Europe - and particularly Eastern Europe - has gained a reputation for a cultural aversion to non-meat dishes, or at least a tendency to prefer non vegetarian food.
Actually, as a vegetarian myself of six years, I found Poland to be one of the easier European countries to find suitable meals in. The national dishes (with the obvious exception of the lengthy Kiełbasa blood sausage, and meat-filled pierogi) are either vegetarian by nature, or can be prepared without any meat at all. The problem is, you have to ask for it.
It's true that most waiters and waitresses in major cities will understand what you mean if you just repeat 'vegetarian' in English, and most menus here will also have vegetarian meals indicated on them; but who wants to struggle like that? Isn't it better to be able to ask for what you want outright? Or, even to be able to request particular dishes without meat, so as not to hinder your enjoyment of Poland's eclectic and interesting cuisine?
Let's start simple. To say I am vegetarian you simply say 'jestem wegetarianinem' if you're a man, and 'Jestem wegetarianką' if you’re a woman (the noun changes to its corresponding gender form with the gender of the speaker); it's the verb 'to be' followed by the noun for vegetarian in the instrumental case (required as you are describing something you act as)
You can also say 'I don't eat meat' (useful for those waiters, and believe me they do exist, who don't really get the concept of vegetarianism), 'nie jem mięsa'. If you’re trying to describe anything else you don't eat, the same sentence starter 'nie jem... ' ('I don't eat...') can be used, followed by the type of food you don't eat. (Remember that the phrase nie jem is a first person negated transitive verb, which means you'll have to apply some specific grammar rules to your nouns and adjectives [here, the genitive plural]). For example, you can say 'nie jem sera' (I don't eat cheese), or, 'nie jem ryb' (I don't eat fish).
There's also many instances where, with the lengthy (and tasty) descriptions of food you will come across in Polish restaurants, you'll be left unsure as to whether something is suitable for vegetarians or not. Here you can ask if something contains meat, or fish, or cheese, simply by using the interrogative form: Czy w tym jest mięso? (Is it with meat?).
Unfortunately there will be instances where it's simply impossible to cook a dish without meat or fish in it, but sometimes this can be done. Ask your waiter or waitress, again with the interrogative form, (can I have this without ham?), bez szynki? (without ham).
Another way of communicating that you don't want to have a certain type of food in your meal, is by simply saying you don't like it; this is less prohibitive and can simply mean you don't like the flavour. This is done by negating the verb 'to like', in conjunction with a sentence object: Nie lubię szynki (I don't like ham).
This is good vocabulary to learn, not just as a vegetarian, but if you don't like particular flavours or perhaps have an allergy to certain foods. What's more, if you have a grasp of the Polish vocabulary needed to get around these situations you'll never be in a situation where you're unsure just what it is you're ordering, and what it contains.