I find that reading music for bass is overrated for anything but pit gigs or high end recording gigs. Not that there's anything wrong with knowing the skill. I cannot sight read on bass, but can use music to help pick out lines. But I've never been given any kind of music and been expected to be able to read it to do my cover band thing - most guys in this arena have no clue how to read music. But, you say you had music training in school and can (used to?) red treble clef. So you have the basic tools in your pocket and just need to learn how to read the bass clef. That's just putting in the work.
Having studied music in school and college, my foundations were firmly set and am essentially a self taught bassist. I did do a lot of surfing and have spent a great deal of time at Scott's Bass Lessons. That being said, I agree that spending some time with the right instructor can save you a ton of time figuring things out and might prevent you from building some bad habits. Assuming you are the type of person who will do the work you need at home, you could get by with minimal sessions with your instructor. Have him/her give you a primer in a specific topic and then go home and work it out. Go back when you're ready for another topic and again do the work. If you're the type of person who needs someone else to keep you accountable for your homework, then set up weekly or bi-weekly lessons.
Regarding learning to walk, it's really pretty simple to do. Take some simple blues songs and practice doing arpeggios over the chord structure. Then listen to someone else working that same song. Try to hear how they are making the arpeggio based lines sound more interesting by adding passing tones or inversions to their version. Go back and start trying some of the things you heard. It takes time, but walking should be pretty simple.
Regarding knowing all the notes, it's good to do so, but for me, I am more of a pattern guy. There are a handful of regularly used bass patterns that are employed in a lot of songs. Once you start seeing these patterns, it's easy to move it from one key to the next. For example, the I-V pattern is the most used pattern we have. There are essentially 2 ways to do it: Root note, down one string and 2 frets to the right (toward the pups). Or root note, up one string same fret. Once you learn this in A, it's easy to do in C (or any other key). I use 7 or 8 patterns (or "boxes" - mostly pentatonic in nature) and so long as I don't use open strings, I can transpose to any key pretty much on the spot.
Soloing is a tough skill for me. All my attempts sound very basic and I don't do much in the way full length solos. Give me a few fills and other short features and I'm fine. I can tell you, that learning the pentatonic scales will be invaluable when start trying to work out solos. I know I'll get flamed for this, but my advice is to not get sucked into "modes" to begin with. They are simple in concept, but can hurt your head for a long time if you get sucked into their depths. Save this for when you have nothing left to learn.
Most of this stuff is not complicated and just tales some time and effort to get it figured out. Good luck and don't sweat it when hit walls (feel like you're going nowhere). It happens to all of us. Sometimes the answer is to keep plugging; other times the trick is to put things down for a few days and let your brain do it's thing. I can't count how many things that were hard for me, just started working after letting it rest for a few days.