A SSD is the most you could upgrade to. It would help with loading some programs but not with others.
The E2 processor in your system is a CPU and GPU that are integrated into one die package, which AMD calls an APU for marketing purposes. The E2-1800 is a Ball Grid Array CPU/APU which is soldered on to your system's motherboard which makes it non-removable. The E1 and E2 lines are possibly the worst series of processors to come out within this last decade, and it's performance levels are on par with processors from 2005. Usually you would find these E2 APU's in cheap laptops that were Black Friday door-buster specials. Laptop parts are also used in entry level All-In_one PC's. This would explain many of your woes.
Some people like to reference Passmark scores which is sometimes a good measuring stick to compare older forms of CPU architecture. Using Passmark as a reference really depends on the question that's being asked. For this question Passmark is useful.
https://www.cpubenchmark.net/cpu.php?cpu=AMD+E2-1800+APU&id=251
https://www.cpubenchmark.net/compare/AMD-E2-1800-APU-vs-AMD-Athlon-II-X2-250-vs-Intel-Pentium-Gold-G5400/251vs136vs3248
Your E2-1800 scores a 818
The Athlon II x2 250 which was an entry level CPU from 2009-2011 gets more than double the Passmark score. The Intel Pentium g5400 which is an entry level CPU from today more than blows both the Athlon and E2 out of the water.
Even if you were a magician with a solder gun and flux, if you managed to install a better BGA413 CPU, the BIOS on your system would probably not recognize it.
The reality of youtube is you do need a little bit of GPU power to load the videos, but you don't need a lot. Modern day Integrated graphics are more than capable of handling Youtube.
A SSD would help the system boot faster and it would help anytime the system is waiting on the hard drive. It wouldn't work for any kind of multitasking. 6gb of RAM is enough for the basic tasks like watching Youtube videos and surfing the web. Even 4gb of RAM is enough for those things.
Right now Solid State drives are the cheapest they've ever been. Last year a 250gb-256gb drive was right around $90-100 and a 500gb-512gb drive was around $150. Now you can find a 250gb drive for $40-60, a 500gb drive is around $85, and a 1tb SSD is closing in on $150. It wouldn't hurt to put a 250gb SSD into that system to see if you wouldn't be happy with those results.
If you aren't happy with those results then you would have to buy another laptop or desktop. If you do end up buying a new desktop, please get a regular sized mATX version. Making upgrades to an All-In_One is next to impossible and upgrading a SFF (Small Form Factor) is very difficult. If you live in a big city then you can always shop on Craigslist for a used PC that someone wants to offload. Sometimes you can find a good tower for $150 that's worthwhile but you'd also have to know what to look for in a PC. You will also find people on craigslist who think the $1500 PC they bought in 2010 is still worth $1200.