There are several common mistakes vegans, often new vegans make. They stick with transition foods as if they are healthy for the long term. Oreo's are vegan but over consumption is very far from ideal. They are perhaps close to the top of the processed food pyramid. A good vegan diet, in my opinion leads you down that whole-food approach. But let's start with vitamins...
My hope is that most people going vegan are aware, but getting Vitamin B12 is a almost a must. You can get B12 from plant sources such as seaweed but they are not necessarily reliable. Also many foods are fortified with B12, so for instance you are drinking plant milks many days then you should be okay. You don't need B12 each and every day since it is something the body stores. In fact if your levels are okay it could take two years to deplete them. But good B12 levels are strongly recommended for longevity. Interestingly about a third of meat eaters are short of B12 as well. Also all B12 is made by bacteria and one of the reasons we don't get B12 from plants anymore is because plants are washed.
It is well known Vitamin D comes from sunshine and that is the most recommended source for humans. However in Winter and as we head away from the equator the quality of sunshine makes good vitamin D difficult to get. Research in the US shows "the overall prevalence rate of vitamin D deficiency was 41.6%, with the highest rate seen in blacks (82.1%), followed by Hispanics (69.2%)." So whatever diet you follow Vitamin D deficiency is a risk factor. Again cheap sources of supplementation are available. Ideally D3, but the body can convert to D2.
Lastly omega-3 (EHA/DHA) is regarded as brain fuel and shortage can be linked to brain fog, depression. It is possible on a low fat diet for the body to convert ALA to EHA/DHA. If in doubt again vegan supplementation is easily available which has none of the downsides of fish oils (heavy metals, rancidity, depleting the oceans etc.)
All other nutrients are plentifully available and accessible to vegans with no research showing actual deficiencies. There are nuances such as to add something like some lemon juice to meals to help improve availability of iron absorption but these are more important to people will illnesses related to anaemia.
For people who follow my posts you will see also the importance of getting sufficient calories. This is easy to do. Simply ask yourself do you have enough energy? if the answer is no then you need to eat some more. Carbohydrates are the source of energy for brain and body.
Don't assume you can go to a fully healthy vegan lifestyle overnight. Phase it in slowly. Remove one non-vegan food as you introduce a vegan one. Doing this week by week will allow your body time to adapt and also give you time to find the best and cheapest local sources for vegan foods such as pulses, legumes etc.
Finally if you have specific health issues then get advice on dietary choices from someone who knows about the diet you want to adopt. Also research as much as you can. There is plenty of information on the web: sadly much of it contradictory. The first step to decipher good info from bad is to check the research. The second step is to not just accept research because it has been published. Check the authors and how the research was funded for bias.