I'd suggest that the Tasche is fine for occasional use (quick note-taking etc.) but, being a pocket-sized pen, it's short, even when the cap is posted, and the section (the bit you hold) is slim, which makes it unsuitable for more than a couple or three lines of writing unless you have small hands like mine - I had one, so I speak from experience. Also, if you're going to get into fountain pens, I would urge you to consider a pen with a converter so you can use bottled ink. Cartridges are fine, of course, but, to me, it's a bit like shaving with a DE razor, but using canned foam rather than lathering a good soap or cream with a brush.
Another thing to consider is which nib size (line width) to choose. You want to avoid loops in "a's" and "e's" becoming blobs due to using a nib that's too broad for the size of your handwriting. Conversely, if your handwriting is large, and you use a fine or extra-fine nib, your writing can seem out of proportion.
My suggestion would be one of the following options:
Lamy Safari from
http://www.cultpens.com, plus Lamy converter from the same retailer totalling £17.92 excluding p&p, though you'd need a bottle of ink on top of that, which will put you just over the £20 mark.
OR
Pilot MR from
http://www.thewritingdesk.co.uk, plus a Pelikan piston converter from the same supplier. This combination puts you a bit further over the £20 mark, but the Pilot is a metal-bodied design, whereas the Lamy is plastic. Again, you'd need bottled ink if you went for the converter.
Then again, with a converter which, with basic maintenance can last years, plus a bottle of ink which will last for months, depending on how heavy your usage is, you'll save money over buying cartridges. Also, there's a vast range of ink colours, as well as variation in ink viscosity (often termed "wetness"), colour shading (lighter and darker tints within your writing) and drying time, enabling you to select the ink that perfectly suits your requirements.
This chap on Youtube
https://www.youtube.com/channel/UCeWDDbfQxKv0Cgq_UNpwYpA is one of the most knowledgeable fountain pen reviewers, IMO, and he has several videos on choosing a fountain pen, including nib options, and I'd recommend spending some time with his videos on choosing a pen, in addition to his reviews of the two pens I've suggested, before making a decision.
Lastly, be warned - if you've already spent more than you planned on shaving gear, getting into fountain pens can be equally expensive!:icon_razz::angel: