Neural network, Support Vector Machine or something else to classify into 7 groups
Welcome to the site, @intellion! I can recommend a couple of things for you. First, if you think you'll continue your interest in data science and will be working on problems like this in the future, it is worthwhile to start familiarizing yourself with the field. There are a variety of introductory textbooks and courses you can choose from. If you have a strong mathematical background, I'd recommend Tibshirani's "The Elements of Statistical Learning", which is a solid classic in the field. If you're approaching this from a more application-oriented angle, I'd recommend something written specifically for the language you'd like to work in. If you plan on using R, there's "Machine Learning with R"; if you plan on using python, there's "Machine Learning in Python." There are of course a variety of free online resources as well.

To your question, you could certainly solve your problem with Support Vector Machines (SVMs) or Neural Networks, but I don't think that's the first place to start. To me, this looks like a classic multiple regression application situation (for an overview for doing this in R, see this or this textbook). If you really want to try something in machine learning, perhaps decision trees are a good place to start, as they'll allow you to easily assess the influence of variables on the final model fit (e.g., using Gini Impurity). I've found the rpart library in R to be quite nice to work with, but it can be a little slow for larger data. You could also use sklearn in python, if that's your preferred language.