- Travelling for business.
Private business restrictions started to be gradually relaxed since promulgation of the law on cooperation in 1988. Prior to that, there was only a very limited possibility for an individual to have a private business in the USSR. In most cases this would be something like shoe repair or knife sharpening booth, with no paid employees - a scale of business too small for the state to bother to create a state-owned enterprise, and for which one definitely doesn't need to travel abroad. In any case, international trade was a strict state monopoly.
There were private or quasi-private enterprises of a slightly larger scale in areas of prestige - such as arts, traditional handicraft, fashion industry etc. - that could justify some international cooperation. These cases were so unique that decisions could be taken on the government level.
One could have been sent on a business trip on behalf of a state-owned enterprise or institution, but I presume such trips are out of scope for this question.
- Studying abroad.
Studies abroad remained available even after revolution. As Stalin tightened the screws, such studies were stopped in early 1930s and many absolvents faced suspicion or direct accusations of spying or subversive activities.
After WW2, a "socialist bloc" emerged, and it was a matter of prestige to let some student exchanges between USSR and other "socialist" countries. One had to be very trusted to get selected into such a program. In late 1950s, exchanges became possible with "capitalist" and "non-aligned" countries, such as the USA. These, of course were even more limited in numbers.
Again, it was a matter of prestige to let some people out of USSR for tourist trips, so some limited travelling was allowed to "socialist" countries after mid-1950s and to "capitalist" and "non-aligned" countries after late 1950s. However, it was tightly controlled and supervised. Tourist vouchers were distributed through trade unions, and one had to be considered fully loyal to be approved by the local CPSU and KGB bodies. The trip was always in an organized group; of course, selection and supervision in "capitalist" countries were much stricter.
The scale of travelling grew gradually. In 1956, there were ~561K tourists. In 1959, just a few hundreds went to "capitalist" and "non-aligned" countries; the population of the USSR was around 200M at that time, so one can get the idea how exceptional this was. In 1980s, the number reached ~5M tourists per year (~80% of them to "socialist bloc" countries).
In fact, in the last decade of the USSR there were even some steps towards liberalization of travelling. For example, USSR had concluded a few treaties with other "socialist bloc" countries on visa-free travelling, which slightly simplified things. Travelling to some of them was made possible even on the "domestic passport" (essentially the ID). One still had to obtain a permit though.
It has to be said that, even though trips were partly subsidised, they were still very expensive. My grandmother undertook a trip to Poland in 1965, and she told me that she had to borrow for it and that it took her a year to repay. So, when a few years later she was offered the opportunity to go to Hungary, she refused. However, some could make some tiny smuggling business out of such a trip and make it profitable.