The Insurrection isn't really one thing as it is a series of insurgencies across human space, with each colony having its own insurgent groups using whatever they have available. CMA equipment was one source of weapons prior to the HCW, but there are groups who performed pirate raids.
Their methods of attack depend on the group and their resources. Eridanus II's rebels attacked places occupied by soldiers, but it was seen as relatively safe for civilians (so long as they didn't get in the way). A character in Hunt the Truth stresses that rebels there did not make whole streets disappear, unlike ONI's propaganda claims.
Mamore however was notoriously bloody. Someone got their hands on a nuclear weapon and killed 2 million people by blowing up a city in 2511. SPARTAN-IIIs had to be sent there in 2537, and B312 took part in a mission in 2552.
The colony of Tribute was known for its sophisticated bomb making factories, advanced enough that UNSC tech designed to detect explosive substances frequently failed to identify compositions. One of its bomb makers actually blew herself up in a restaurant in Casbah in 2524, killing over 30 people when she got found out.
On Reach, rebels (possibly non-citizens) seem more crude. In the National Holiday disaster, two VBIEDs were crashed into a cruise ship with no strategic value whatsoever, killing 1500+ civilians.