Shooting down anything other than civilian shipping is dangerous, and civvies tend to be worth negative points and money anyway, also earliest weapons are not going to cut it against the factions. Research gives lots of infamy and infamy is actually worth money, so it's worth putting resources into your brainers pretty quickly, I usually try to hire them at the beginning of a month, as they are expensive to maintain and I want to get research out of them before having to pay wages.
Following people and looking for targets of opportunity is a good early way to find infamy, as are the missions that pop up. Early pogroms are actually not great for your infamy, as you probably don't have the resources to keep civilians alive as well as your crew.
I can't emphasize enough that research is probably the most important source of infamy early on, it directly gives it and provides more ways to gain it as well as more tools to survive getting it.