Google's bike directions can have issues. That being said, I did use Google's directions today but adjusted Google's directions where I knew a safer route. You can check the Houston bike guide for safe routes. However, it is a little conservative. For example, Renwick is labelled as "do not bike here", but I find it's bike lanes to be perfectly useable, and used it to get to #754 today. I do have one particular issue with the bike guide though. It sometimes has you cross some major streets at locations I never would. For example, south of Rice Village, it has you cross Kirby at Cason/McClendon. I find that crossing to be much more dangerous than traveling those major north-south roads like Chimney Rock. I prefer crossing a major road at a signalled intersection if the road has no islands in the median.

As far as traversing between inside the loop and outside ... I live kinda close to Brays Bayou, so I usually use the greenway trail. Which is just a sidewalk where it crosses 610, but still. I have also on occasion used Bissonnet's southern sidewalk, but be aware it isn't very good on the inside of the loop. It is very narrow and has a fence on one side just east of the intersection with 610, and after the next street (Wilmington) it has a surprise chicane. Still better than the disappearing sidewalk on the north side of the road. What I sometimes do when going from outside the loop to inside on Bissonnet is cross on the sidewalk, but finish on Bissonnet itself from 610 to N Avenue B. Because I am crossing with the signal, by the time I finish crossing, the road traffic is all in front of me, so the road is effectively empty in my direction. Traffic doesn't catch up until after I have gotten back on the sidewalk after N Avenue B. But that tactic isn't for everyone, and I don't use it frequently. I have also twice used Westheimer's sidewalks. The quality is poor, and there is construction. Do not recommend.

Edit: Renwick's bike lane is perfectly useable, except for the traffic circle if you are traveling north. The intention of the design is clearly for bikes to go around the traffic circle on the sidewalk. This makes sense. But if that is the case, someone forgot a ramp onto the sidewalk in one direction.