Hey Dup,
I'm a guitar teacher, I've been teaching for over 5 years and am currently trying to build up my students to do it full time.
The first thing I will say is £12 per hour is very cheap, I charge £16 p/hour which is still very cheap (average for my locale - Yorkshire, is 18-20, I would charge more if I had a better room to teach from.) This isn't supposed to mean that he isn't a good teacher, but you get what you pay for usually.
A teacher's job first and foremost should be to help you achieve your goals, the advantage of having a teacher will be that he should help you do this by saving you time, this can be done by choosing appropriate songs/riffs/guitar parts for you to learn that facilitate the topics he is teaching you. What are you wanting to do on the guitar? Do you want to be able to play some of your favourite music? This is the most common reason people go for lessons, if this is correct, then the topics you are covering in lesson should be related to the music you are interested in. If he hasn't asked you questions like; who are your favourite bands, do you have a favourite guitarist, what music do you like, why do you want to play guitar, then how can he know what to teach you? If he has asked you these things then he is considering your goals, which is good.
For example, one of my students loves pop music, he isn't into the guitar as such, meaning he doesn't have a favourite guitar player/riff/solo etc, he likes playing along to music on the radio, such as Michael Buble, James Morrison ( a guitar player but not famous for his playing ), so our lessons are geared towards learning chord shapes and how to strum, so ultimately he can play along with the music he enjoys. Use this example to ask yourself if the teacher is teaching you appropriately, if I was teaching this student things such as string bending, speed playing, tapping, etc, then the lessons wouldn't be appropriate - note there are always exceptions to this 'rule', over time a teacher should introduce you to music you haven't heard, techniques that are new to you, even guitar effects/settings you aren't familiar with, all with a view to broaden your knowledge of the subject and inspire you.
That being said, to be able to play anything on the guitar you do need a basic understanding and grasp of technique, so I will teach everyone the correct way(s) to sit with the guitar, hold the pick, correct left hand placement to fret notes and chord efficiently, then always a collection of exercises involving 'alternate picking' (picking in both directions with the guitar pick), strumming and gradually introduce other techniques; sliding, hammer-ons/pull-offs etc etc (always depending on their own goals). - So I would expect your teacher to have shown you some of these OR you should be working on at least ONE song, as to be honest NO ONE wants to sit and play exercises, he should get you playing a song as soon as possible (I always have the student leave their first lesson with a song to play).
One final thing I will add is make sure you get along with your tutor, a guitar tutor is an investment, and an expensive one, guitar is also usually very popular with a few teachers in a region, you should shop around and try other tutors, don't feel pressured to stay with one if you don't feel comfortable.
In terms of modelling amps, the Line 6 spider amps are decent value, Vox also do some good modelling amps. I haven't heard anything about the Peavey's so can't comment, Line 6 are the most well known for modelling with Vox's amps also being well reviewed, so I would investigate these.
