It's complicated.
apt-get upgrade will upgrade existing packages but not install new ones or remove existing ones. Apt-get dist-upgrade will install or remove packages if needed to upgrade other packages.
The bootloader will normally boot the newest kernel you have installed by default.
There are essentially 3 types of kernel upgrade that can happen within a ubuntu lts release.
- Updates where the kernel package names and (hopefully) the kernel ABI remain the same. These will happen with apt-get upgrade.
- Updates where the kernel ABI has changed (unfortunately some bugfixes affect the kernel ABI) but still within the same upstream release series. These will normally be pulled in by apt-get dist-upgrade but not apt-get upgrade.
- Hardware enablement kernels. These are essentially backports of the kernel for a newer releae. They will only be pulled in if you have the relavent hardware enablement metapackage installed.
Whether you have the hardware enablement metapackage installed depends on the installation media you used (see Sylvain's answer for details).