How to update the Linux kernel version for Debian Jessie?

Question

linux

I want to install docker, but when I start the daemon, it swears that the kernel is too old.

Your Linux kernel version 3.2.0-4-amd64 is not supported for running docker. Please upgrade your kernel to 3.10.0 or newer.

See the current version:

uname -r
3.2.0-4-amd64

I want to upgrade to 3.16.0-6 (the latest LTS version, also available in the Debian Jessie repository).
I do:

apt-get install linux-image-3.16.0-6-amd64 linux-headers-3.16.0-6-amd64

The output of uname -r is the same.
Questions:
1. How can I update the kernel version so that uname -r gives a different number?
2. Does it make sense to use ukuu?
3. What is the best thing to do to minimize breakage during updates?
UPDATE
There were originally two systems on the same machine, both Debian 8, differing only in minor versions. I forgot about the second system, because I did not use it due to the fact that Wi-Fi did not work on it.
On the advice, I updated grub, which was on the partition of the second system:

update-grub /путь/к/корню/второй/системы/boot/grub/grub.cfg

after which, in the rough, the option of launching the first system with a new kernel appeared.
After starting the main system with the new kernel, everything booted successfully, visually the logs looked the same.
The Wi-Fi stopped working.
Checked, the second system initially used a newer kernel.
I have no idea why the Wi-Fi adapter driver or whatever doesn't work with the new kernel, but the pattern is clear.
Now booted from the old kernel, no problems.

Answer 1

Installation is not enough. You also need to reboot into the new kernel. For this, the appropriate keys for grub must be generated, but usually this happens automatically.
2. No
3. If this is the only machine and there are no specific services - such as nfs, ceph, iscsi - then the probability of breaking something is minimal. If you do it smartly, then you need to make a clone of the machine and try to update on it as in a "sandbox"

Answer 2

Did you reboot after installing the new kernel?