If you're only interested in whether there's a difference in proportion of the two types of cancers between the two groups, then a chi-squared test or Fisher's exact test are appropriate and simpler. A logistic regression is possible but it would evaluate the cancer type as a function of the year, i.e. it would answer whether the year predicts the type of cancer, which is not really what you seem to be asking from the data. If you don't know why you should do one approach over the other, I suggest you ask your supervisor for explanation, maybe there are more factors that need to be taken into account.
Anyway, you can do logistic regression with the glm() function in R, something like

model <- glm(cbind(breast,lung) ~ year, data = your.data.set, family = binomial)