Allright I'm basing it off of this: https://tvtropes.org/pmwiki/pmwiki.php/UsefulNotes/Airships
Unlike what others have said, airships are actually exceptionally durable, rule of thumb for storms it can handle is an average speed the same as the maximum speed of the ship and the airship is one of the only vehicles that benefits from the square cube law. The bigger it is the more liftinggas it has and the less surface area compared to volume it has for the wind to push it. Even the WWII airships could reach maximum speeds of 130 to 150km/h and hybrid airships can reach a whopping 225km/h. Consider the beaufort windscale (https://en.m.wikipedia.org/wiki/Beaufort_scale) where violent storm winds (windspeed 11) reach 117km/h and hurricane force (windspeed 12 the highest) is 118km/h and higher and you can fly in most weather even if you had a WWII airship! Here's a piece from the link:
"In the same way, some hot airships are blown backwards in a light breeze, but on the other hand, in the 1950s an American naval blimp program manned an Arctic airborne early warning station continuously for 10 days in the winter. The weather was the area's worst in years, grounding all other military and commercial airplanes with combinations and variations of ice, snow, rain, fog, and 65 mph (104km/h) winds. The airships went on sorties that lasted for days, in extremely heavy icing conditions, and succeeded in their mission with no crashes- effectively crushing the misconception that airships are inherently more vulnerable to weather than other aircraft."
Even with hydrogen airships remain highly resistant to crashes. During the first world war it took entire drums of incenendary ammo to light an airship on fire. Today a small airship can take 500 bullets and 2 hours before there is a danger of crashing, larger one's can take thousands of holes. And the Hindenburg took a while to become engulfed in flames. It took long enough that most of its occupants could jump out and escape during the (crash) landing sequence.
Most accidents with airships in storms happened because of human error such as overconfidence in the airship during storms causing the pilot to brave one they shouldnt have.
Now to the actual answer, what problems would you expect?
flying and especially take-off and landing is hard. Airships fly more like seaworthy ships than aircraft. Pilots were often sea captains for this reason.
you need prepared sites to land, or have special towers to hook on to. Hybrid airships can land more conventionally and some modern technology lets you land airships on the ground but I expect these to still require teams, not a single dude doing a soul-searching world trip. Hybrid airships are likely your best bet as they require the least equipment to land and only require a short runway to land or take off.
cost. Airships are relatively cheaper than similar size aircraft, relative is the key word here. This isnt a cheap old car where if the engine stalls you can pull over and see what's wrong. You would have to do a lot of maintenance and care to make sure your airship stays safe and operational. If airships somehow became widely used and cheap you could reduce this problem.
I actually had a similar idea for airships but thought of using them as superyachts. The fact that no multi-millionair has put exorbitant money into one just to show off a luxurious moveable home is a mystery to me.