Fingertip care is not anywhere near as onerous as caring for nails, but it still needs to be considered. If you have just cut off your nails, be aware that the skin of your fingertip may be sensitive. I would advise using low tension strings and/or tuning your guitar down a semi-tone or even a whole-tone. Gradually increase the tension of the strings to normal tension (if you want to play at 440hz, that is) so that you ease your fingertips into their new role without damaging them. This can take months.
I wrote about fingertip textures here. The worst cause of rough, ugly-sounding calluses is blisters. These are easily avoided. If you have been playing without nails for a while, your skin will likely have become firmer and more resistant, which will help. I would advise not playing immediately after showering or bathing — your extra-soft skin may feel lovely to play on, but it is especially vulnerable to blistering. Also, moisturise your fingertips — I usually do it after playing. Taking breaks when practising can be useful (as it can for many other reasons). At this point I can practice for hours without an issue, but when I started out without nails I had to be much more careful. (Which I learnt the hard way, after getting a horrid, ugly blood blister that took weeks to heal.)
If you do somehow manage to get a blister and have, for instance, a performance coming up, a quick fix is to file down the dead skin with a glass file or fine grit sandpaper (avoid anything more abrasive, like an emery board). Some players even do this regularly, rather like polishing nails. For me, however, filing the skin makes it grow back in a less-than-ideal way — it fixes a short-term problem but creates a long-term one. I would only recommend it as a last resort. Like with every problem, prevention is the first and best solution. Take care, take breaks, and moisturise.
Also, while we may not have to worry about breaking a nail, we can cut the skin. I did this once when chopping up food. Catastrophe: I was completely unable to play until it healed. Even a little cut can be a big problem.

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