Advice on typing Russian and Ukrainian
Most West-European languages like English, French and German use the Latin alphabet. East-Slavic language such as Russian and Ukrainian don't. Instead they use the Cyrillic script. In the same way that there are small differences between the French and English alphabets, the Russian and Ukrainian alphabets differ from each other as well. For example, the Ukrainian letters Ґ, Є, І and Ї don't occur in Russian. Planning on learning Russian and Ukrainian myself, I'll share what I learned so far about the different ways you can type in Russian or Ukrainian on a computer.
Phonetic keyboard layout
There are many similarities between the Slavic and Latin scripts. For example, Cyrillic М is very similar to Roman M in appearance and sound. Other letters such as Д look less similar, but this letter has the same sound as an English D. If you use the standard keyboard layouts from each country, then these differences are ignored, but a phonetic keyboard layout maps the Cyrillic letters to their most logical* Roman equivalent. (*There are more symbols in the Cyrillic script than in the Roman one and different phonetic keyboard layouts handle these extra symbols differently).
Slavonic Multilingual Phonetic (for Russian and Ukrainian)
A phonetic layout I like myself is the Slavonic Multilingual Phonetic keyboard layout. This layout is optimized for Russian, but can produce Ukrainian symbols that don't occur in Russian by holding done the Alt(Gr) button on the right of your keyboard.
There are many alternative phonetic keyboard layouts, but most of them focus on a single language. You could use these if you're planning on learning only one of them. If you want to learn both Russian and Ukrainian, pick the multilingual one above. If you're using MacOS or a Linux distro, it's likely that you already have phonetic keyboard layouts pre-installed for Russian and Ukrainian.
Google Input Tools
An even easier way to use the Cyrillic alphabet is Google Input Tools. This lets you type using the Latin alphabet and brings up a list of suggested words to select from. The good thing about this method is that it works as well for other scripts, such as Chinese and Japanese. The disadvantage is that you get less practice in learning how to spell.
Using the official keyboard layouts
If you're planning on moving to a Russian or Ukrainian speaking country this might be the most useful layout to learn; however, this option has a much steeper learning curve than the others.
Unless a given individual is going to have to use a standard Russian keyboard, I strongly recommend the phonetic one if it's available for your device. I had to use the standard one and currently only have that available to me, and despite that I still find one that is mapped to equivalent qwerty letters is soooo much more intuitive to use. Really not a fan of trying to use the standard one, it sloooooows me down so much. And at the moment my Russian is already pretty slow ;p not a fan of the йцукен layout, it's a pain in the backside!
That multilingual Slavic one is very neat.
For Linux, combined RU+UA layout is available out of the box. It doesn't cover all symbols you've put to that layout, but on the other hand it is readily available on (almost) any linux install and is based on standard ЙЦУКЕН layout.
EDIT: you've mentioned this layout yourself.
For Android users, I highly recommend a somewhat strange keyboard called MessagEase. I use it for English, Spanish, French, Cyrillic and Ukranian, and it has many other keyboards. I haven't learned to touch type in Ukranian yet, but with this keyboard I don't really need to. I have been using this keyboard with Duolingo for more than a year and it is great. It is available in the Play Store and, amazingly, it is free. To use it fluently it has a somewhat difficult learning curve, but I think it might work well fairly quickly in hunt-and-peck style.
I use the Microsoft Russian keyboard and the keyboard layout is completely different from English or the one you show above, but I already learned it. Oh, you do have a link to Microsoft Russian keyboard layout. I think the keyboard you show above is closer to the English format with letters that sound like the English ones in the same places where possible. I am definitely interested as I will want to learn Ukrainian later. I am just worried about switching keyboards at this point. I have to check if the Microsoft Ukrainian keyboard is similar enough to what I already know. Whoa, here I go again. If I had it to do over, I think I would have used yours.
You're probably talking about this layout img, which is what the Russians themselves use. The idea of a phonetic keyboard is to be as close as possible to the English format to simplify the learning process.
The standard Ukrainian keyboard doesn't look such different img, but it's still a nuisance to have more than two keyboards installed.
Not all English letters have a Russian equivalent, either.
The ones that do are mapped appropriately, the ones that don't have equivalents are on the leftover keys, plus I think the твёрдый знак and the мягкий знак take up a couple of spaces, etc... because Russian has more letters, you lose a little of the punctuation that you usually have instantly to hand on a Latin keyboard.
Overall, it's surprisingly intuitive and easy to get used to, and I much prefer it to the regular йцукен keyboard, even though I have at least as much (maybe more) practice typing with the latter. The phonetic keyboard, once you have got your head about where the extra letters turn up, is, IMO, way easier for someone who is used to typing on a qwerty keyboard than the standard Russian one is.
It's also easier to use if you don't have the Russian layout on your keyboard. It is possible to get a 'billingual' keyboard, ie one where both alphabets appear on the keys in the standard layouts, or to purchase stickers to add to a regular keyboard with the Rusian letters on, but personally I still prefer the phonetic keyboard over either of those approaches, and it's also, I think, easier if for whatever reason you can't use or create a keyboard with the Russian letters on it. It's much easier just to memorise a few letters for a few keys and be able to type the rest more or less intuitively because they're mapped to a layout with which you're alreayd familiar.
Additional letters are usually available if you press and hold a usual "key".
For example, on my Android phone ALL Ukrainian letters are available on a Russian keyboard, hidden in Э, И, Г (Russian does not use Ґ, І, Ї, or Є). I am sure, it is the same on a Ukrainian keyboard: ї might be hidden under і to save space.