Trails are in good shape, most places are that good balance of wet enough to be tacky but not sloppy.
That map is the right map, but pick up a copy (they used to be $5) at one of the local shops to support the trail builders. Look for Outdoor Addictions or Swicked at the north end of town. There are also ride suggestions on the map.
Family ride you'd probably want Riley Lake, Lost Frog or Lost Lake. Mudhoney is fun but does have a few more techy sections that beginner riders might not like as much. You could park at the Lost Lake parking lot, go up to Tri Mac, Riley Lake, then Lost Lake looping counter clockwise to the Tri Mac turnoff. For more techy riding, park at the first lot on your left after the dam, take Alligator Shoes to the road, follow the road left for 100m and go right onto Upper Deliverance. When you finish that, go right on Mudhoney and then a quick left onto Sanitarium. That will drop you onto Lost Lake. Go right on Lost Lake, and then right again on Mudhoney. Follow Mudhoney until you see a trail on the left called Grilled Cheese, follow that for a minute then right onto Stickman. Stickman will spit you out on the road across from Alligator Rock. Go right, but this time turn left onto Lower Deliverance. Stick to the left at the first intersection, then you'll be on Alligator Rock heading back to the parking lot. That's about 10k, and it's a fun techy loop. If you want something longer, there are lots of trails that can be added to that loop. Follow the suggested direction arrows on the map and you'll be ok.
Being cheap is OK. Being a clueless sanctimonious condescending douchebag is just Vlad's MO.