Update Upgrading the kernel? Moving to 24.04?

I have a question in 3 folds regarding Ubuntu releases vs. kernel vs. my laptop and the actual installation.

Disclaimer, please read carefully: I've used Linux for a while on this laptop, but I know little to nothing about Ubuntu/Linux since I've only been using Brave and LibreOffice for 99.9% of the time. Please don't expect me to have even a basic knowledge of all this. I'm doing the best that I can.

My problems

I downloaded and installed Xubuntu a while ago, then sudo apt-get update and sudo apt-get upgrade regularly, however it seems that for some reason my kernel is completely wrong. I cannot activate Live Update because it gives me an error saying it can't because of the kernel I have at the moment. Also, my laptop has been sluggish since day one and the internet speed is awfully slow (wifi and LAN). I have a desktop on Windows here and the net speed is fine so the problem is 100% on my laptop for sure. The kernel has to be fixed and if it doesn't fix my speed as well I'll post another question regarding my drivers and settings (mtu, etc..)

Actual install

hostnamectl Static hostname: xxxxxxx       Icon name: computer-laptop         Chassis: laptop      Machine ID: xxxxxxxxxxxxxxxxxxxx         Boot ID: xxxxxxxxxxxxxxxxxxxxOperating System: Ubuntu 22.04.5 LTS                        Kernel: Linux 5.19.0-32-generic    Architecture: x86-64cat /proc/versionLinux version 5.19.0-32-generic (buildd@lcy02-amd64-026) (x86_64-linux-gnu-gcc (Ubuntu 11.3.0-1ubuntu1~22.04) 11.3.0, GNU ld (GNU Binutils for Ubuntu) 2.38) #33~22.04.1-Ubuntu SMP PREEMPT_DYNAMIC Mon Jan 30 17:03:34 UTC 2

1. Kernel release

What is the latest kernel release that fits with 22.04 jammy? I searched a LOT on the web and numbers go from 5.x to 6.2 to 6.8, etc. I also checked at https://kernel.ubuntu.com/mainline/ and the last entry is v6.13-rc3 but is it the best one to use? Or is 6.12 better with 22.04 jammy? (or something else).

2. Upgrading the kernel

Is it possible to easily upgrade my actual kernel 5.19.0-32-generic to the latest best one? If so, can anyone provide me with a step-by-step, or redirect me to a webpage with such information? Again, I searched a lot, I saw many different ways of upgrading the kernel, and I have no clue which one is best and the safest.

However, I've found this method that seems to be good:

1. sudo apt-get update
2. sudo apt-get dist-upgrade
3. If the above doesn't work, then do this: update-manager -d

I've also read this:if you'd like to install the latest stable kernel, you can simply runsudo ubuntu-mainline-kernel.sh -i

3. Stable release

https://ubuntu.com/about/release-cycle shows 24.04 LTS Noble Numbat. Should I just save everything and install 24.04 instead of upgrading my actual kernel? Here's my laptop:

   CPU: AMD E-240 1.5MHz - 64-bit
   Memory: 5760MiB
   VGA: integrated graphics (AMD Wrestler Radeon HD 6310)

A big thanks to @BeastOfCaerbannog who edited a previous post of mine without giving me xxx LOL. I've learned a better way of using code to make my question more "readable".

Update 2025-01-12-22:06

I did as instructed and it failed. I even pressed the Tab key after just in case and nothing happened.

sudo apt install linux-image-[sudo] password for moi: Reading package lists... DoneBuilding dependency tree... DoneReading state information... DoneVirtual packages like 'linux-image' can't be removed0 upgraded, 0 newly installed, 0 to remove and 46 not upgraded.