It took me 11 years to get to 328; 338 took 19 years.
Yes, I was hooked on DXing within weeks of getting my novice ticket. I focused on operating in CW so I could pass the 21 wpm extra exam and gain access to the low ends of the bands, which took about 6 months.
With good propagation knowledge, operating skills, equipment, and software, working the DXCC entities with resident hams can be accomplished in a year or three. Working the rest requires DXPeditions, and all that they entail.
73,
Dave, AA6YQ