Is it permissible to search the internet for news of old friends and acquaintances and the changes in their lives, or is that considered spying and tracking private matters?

If the news is apparent to people, like marriage and the like, then your inquiry about it is not considered prying into private matters. In fact, you might even be rewarded for it with a good intention, such as strengthening kinship ties and love for the sake of Allah. However, as for news that people are keen to keep private, it is not permissible to pursue it, due to the general meaning of His Almighty's saying: "And do not spy," and this is considered prying into private matters which is forbidden. This is further indicated by the Prophet's (peace be upon him) saying: "From the excellence of a person's Islam is his leaving alone that which does not concern him."