Are 3D prints safe to touch right after printing? If not should I do some kind of curing (like in resin printing) when my prints are done or wash them with alcohol or something.
FDM/FFF runs by melting a filament and letting it cool again. As soon as the printed parts are cooled down to below 50°C (which is rather quick) handling them is safe. There's no chemical process involved, it is a pure physical, thermoplastic process, and the plastic re-solidifies as the part is cooled down.
It is far more likely to get damage from touching the hotend than from the printed part.
SLS/DLMS aka Fused Powder prints: Yes but...
SLS/DLMS runs by pointing a high-powered laser on a fine powder and melting/smelting/welding/sintering it together. While such powder prints, no matter what type of powder they are made from, are safe for your hands to touch, the fine powders are very bad for your lungs. Handling such prints requires breathing protection until they are dusted off, usually using compressed air.
For stability, some are impregnated with resins after print, which requires hand protection till the resins are cured.
DLP/SLA aka "Resin": NO
Resin parts are made by curing a liquid resin.
The most common type is DLP/SLA, where a laser or light source cures the resin at the bottom of a vat and the model is pulled up from. As a result, they come coated in liquid, uncured resin. These resins are a known skin irritant, and their fumes can be an irritant too. In general, keep them in a dedicated workshop.
Because of this, these prints are not safe to handle barehanded and need to be washed and cured after printing.
Polyjet Resin Printing: kindoff
There's a method of resin printing that is more akin to an inkjet printer, such as the Stratasys Polyjet system. Those prints don't come coated in resin and come out cured to a point where handling them is safe, but those are (as of 2022) industrial machines, and even as the PolyJet core patent expired in 2019, it is a market that nobody jumped at, especially considering many of the advanced features like full-color Polyjet using CMYB is under patent still. Oh, and apparently, even in 2022, only one other company managed to make a functional Polyjet machine that is ripe for the market, which is super tiny in the first place.
Back to topic. While the parts come out fully cured, polyjet printing itself is messy. You absolutely need support material everywhere. It is safe to touch, but in the dissolution stage of the support material, it is most certainly not safe to touch.