Been vegetarian for around 30 years. My guess is that if you're not sure, then of course you're not getting all the nutrients you should. "Need" is relative - you can survive without them, but some nutrients that are easy to miss as a vegetarian pretty clearly lengthen life expectancy, reduce development of certain diseases, and influence neurological function.
The most common thing uninformed people worry about for vegfags is getting enough protein, or the right balance of amino acids, which in actuality is almost never an issue.
I would look into omega 3 fatty acids. That's one that isn't even listed on many packages, since the FDA doesn't consider it an "essential" nutrient to sustain life, even though its effects are widespread and significant. Flax seed or flax seed oil are classic go-to foods for vegetarians concerned about it (just add a tablespoon of oil a day to something you eat that calls for oil)...walnuts and olive oil are inferior also-rans, but help a bit, though beware that walnuts cost an assload ($10/pound at my grocer).
Vitamin D can be an issue for any diet, though egg yolks, and fortified dairy & grain, are common non-sunlight sources; vegans are more at risk than vegetarians.
Vitamin B12 is less of an issue for meat eaters, but for vegetarians is gotten more through fortified dairy and eggs, making it another particularly concern for vegans, but something anyone should keep an eye on. A daily multivitamin can keep you covered.
Calcium is another concern for any diet, but of more concern if you don't consume much dairy. Some veg foods have some, and enriched bread or faux milks help, but boosting through supplements. (Vegan calcium supplements typically cost more than non-vegan).
Minerals like zinc and iron can also be an issue. You can get iron from cooking in a cast iron pan, and lots of enriched flour products include these, but you might still be short. Use iron supplements if you're a frequent blood donor.
