#1
I started taking voice lessons recently, I've always wanted to be a good singer. I've been playing guitar for a decent amount of time, and some other instruments. I'm okay with where I'm at guitar-wise, but I'm looking to become at least a competent singer. I'm a total beginner now. I've tried to learn through websites, but it's really difficult.

Anyways, I'm concerned I may have chosen the wrong teacher. I get along with him well, he seems pretty musically knowledgable. He teaches at the local music store. But I feel like this week was a waste, and nothing will get accomplished. I've been taking lessons for 2 weeks. Here's how it went:

Week 1: Introduce each other, we sing along together to some songs I picked, he says to do the lip roll exercise.

So, that's a typical first week lesson anywhere. We didn't really accomplish much, I figure he wanted to see where I'm at vocally. So we sang a couple John Frusciante songs (I'm a big fan of his). He talked about how JF switches between voice registers. And then we started to work through Hazey Jane I by Nick Drake, but ran out of time. He was figuring it out on piano, chords, melody, etc, but we didn't sing it.

One thing that worried me is I felt time was being wasted by him figuring out the songs on the piano (didn't take too long, but still 30 mins isn't that long, so I want to get as much done as possible)

So my assignment for that week was to do the lip roll exercise, and sing along to the Frusciante song (the parts I was comfortable to sing, not the real high notes he does). So I did this. My worry is we didn't really discuss proper technique, like breathe through the diaphragm, resonance, tension, etc. We just sang along to the song, I feel like we should have talked about technique more.

Week 2 went like this:
Do you have any songs you wanted to sing today? Me: Umm (he didn't say to pick songs to learn last week). So I was like how about 12:51 by the Strokes? I figured it would be a good start, the guitar plays the vocal melody for the most part, and it's in a comfortable range for me to sing. So we sing that. He said to practice singing to a guitar. And that's really all we accomplished. He said I need to sing with more air, and to be more confident singing. But again, no mention of proper technique, or exercises yet. I don't really feel like we accomplished anything that lesson.

My assignment this week is to sing 12:51. But keep in mind, I'm a total beginner at singing. So I feel this week is going to be a waste of time. How will I get better? I'm just going to sing a song, when I don't know how to sing properly. It feels like I'm just practicing bad habits. I feel like we should work on proper technique before I start singing songs.

I think of it like guitar lessons. My guitar teacher was a really good teacher. He'd show me a certain technique/chord/scale, and then give me a piece using that technique to practice. So I'd learn the techniques, and actively use them. He was strict about making sure I'm playing the guitar with good technique too. And this teacher just tells me to sing the songs. To put it in guitar lesson terms, he's giving me songs to play, but he hasn't showed me how to play chords yet, I feel.

I'm going to bring up my concerns to him next week, but what do you more experienced singers think about my situation? My concern is that we don't work on technique enough. Songs are great, but I want to sing them with good technique, and right now he hasn't showed me good technique, he just says to sing songs. Am I right to be concerned? Sorry for the long post.

tl;dr: Took voice lessons, teacher has me sing songs, hasn't gone in enough detail about proper technique/hasn't given me exercises to improve it.
#2
I would wait for a while
you cant teach someone to control their voice you can only give advice and plus it should be totally natural, technique can come later

think of it like if a guitar teacher showed you who to play chords before he taught you how to tune your instrument

give it time and if you want bring it up to him and see what he says
#3
I'd move on.

*Everything* in singing is dependent upon technique. Everything.

Go back through your post and switch the words "singing" to "playing guitar."

He told you to "play guitar" along with the recording and to "play guitar" along with him. But did he show you HOW to do it? No. Was there any indication that he was going to show you HOW to do it? Not apparently.

With me, my first lesson is a run-down of the four stages of the technique and how it works, and what instruction will look like as a means towards developing that technique.

Take that as a point of comparison.

CT
Could I get some more talent in the monitors, please?

I know it sounds crazy, but try to learn to inhale your voice. www.thebelcantotechnique.com

Chris is the king of relating music things to other objects in real life.
#4
I would give it another week and see what happens. But when I started to take vocal lessons, right out of the gate we started with technique. She heard me sing one song, then after that she immediately started to tell me what I was doing "wrong" and/or what I could improve on and she gave me a few breathing exercises. Every time you sing, or at least most of the time, he should be breaking down your performance, what was week and what was strong so you can learn. So, if this week he doesn't start you with technique stuff and exercises, I would start to look for another teacher.