Instead of the forced tutorial, I think it would be better to make the gameplay more intuitive. Make it so you get the upgrades by shooting them and you stop the bombs by shooting them. Though it would make more sense to collect upgrades by *not* shooting them.
The second gun really doesn't feel like an upgrade. Sure you might occasionally hit two enemies at once, but if it takes twice as long to fire, it's just not worth it. One thing you could do is make it so they reload independently, so even if it's not as good it's worth using. Or maybe you have the main gun you fire by holding the mouse button, and then press buttons to fire the other guns, and they each have their own icon showing how long they take to recharge.