Is this normal or am I doing something wrong?
It's normal, 3 independent colour systems one based on light, one based on 4 plates merging to form a single colour on paper (Cyan, Magenta, Yellow, Black) and one that mixes inks to form a specific colour (against swatch registration samples) onto paper. Bear in mind that even within a single medium like print, CMYK (the common approach) cannot create every colour in a design - it will struggle with orange for example. If orange is a key part of your presentation or the client has asked you to match a specific brand colour, my routine is to go CMYK plus a 5th (Pantone) to ensure the colour they want. Clearly this is more work though as you have to separate the colour in your artwork for it to work on press. It's a good skill to learn though as it can be used to add metallics and finishes like UV over your printed image.
How do I correctly convert that into an RGB value?
There are several ways to do this. The general rule of thumb is to take control yourself, research/trial a routine that you are happy with. Don't rely on software like Acrobat to convert for you... this puts the conversion into the hands of a programmer (albeit a good one) and it may be disappointing.
Should I use the RGB values Illustrator gives or the ones color picker in Photoshop gives?
The point made above about Adobe Bridge is good in respect of removing as many instances of conversion as possible i.e. create a profile and then make sure all Adobe apps stick to this one. I would extend this to include "use one programme primarily and consistently". I find Photoshop quick and easy due to the picker - you can sample a colour (or choose a Pantone value) and get the closest Hex, RGB, CMYK quickly with the CAVEAT discussed above... it wont work in all circumstances and some colours are particularly difficult so in practice you are best to have a second 'checkpoint' for each medium e.g. RAL chips for physical paint matching (for exhibition stands / office interiors), Pantone books for brand colours in print etc.
One valuable tip that has served me well for 18 years. If I am designing the brand from inception, I deliberately choose colours that match closely across CMYK and Screen even though I will give the client the appropriate Pantone references in the brand guide I deliver. Why? Because this makes my fulfilment role much easier and often I can save time / add profit by avoiding expensive 5th and 6th inks printing everything in CMYK (or increasingly on an Indigo digital press) with a decent match to brand. Purists may call this bad practice, a shortcut and suggest that Pantone should always be used for branded print but we live in the real world. (a) 95% of clients wont notice / don't care. (b) I need to make profit from the job as a commercial designer... If the client is corporate, high street recognised, global and can afford detail, then I will put time in the job for special colours, press checks, print on stock samples etc.