These days, you go for a Ryzen.
Not me. And not a lot of other people either. Ryzens are good processors but they are not the answer for everyone.
Something else to consider is a new motherboard is considered a new computer in terms of software licensing. Not all software licenses are legally transferable to new computers. This is primarily true of OEM software - software that came pre-installed on computers, or was bought for a specific computer. This most commonly includes Windows licenses. If your current license is an OEM/System builder's license that came with or was purchased for that computer (and since it appears to be a HP computer, it most likely is an OEM license), you cannot legally use it with a new motherboard. Note this includes upgrades (like W10) that was authorized based on an OEM W7 or W8.1 license.
So if you do get a new motherboard, you will most likely need to buy a new Windows license too - unless you want to be using Windows illegally.
Also, you didn't post the board/computer specs so we can't be sure but there is a good chance your current motherboard supports DDR3 RAM. Most new boards support the newer and better type RAM, DDR4. So if you buy a new board, you likely need to buy new RAM too.
So while Intels tend to cost more, a new Intel will probably cost considerably less than a new motherboard, new RAM, new AMD processor, and a new Windows license too.
Since you are starting out now with a dual core i3, a nice quad i5 will provide a very significant and enjoyable performance boost without breaking the bank.