It is proven practice not to optimize programs before you don't have
- indications for a noteable performance problem
- you know for sure the place where you are optimizing is really the root cause of that problem
Said that, if you have an indication that at least one of the lookup values is causing performance losses, it could be simple in your case to provide a generic solution which provides a general preloading strategy for all lookup values (maybe simpler than providing an individual strategy for just one value).
If that is the situation, implementing such a preloading may be fine. Be aware, this does not come for free: preloading hardcodes the assumption into your program that those lookup value never ever change during the life time of your application process, and if that assumption becomes wrong, this will cause additional development effort to mitigate this.
This is well known as cache invalidation, and often mentioned as one of the two hardest problems in Computer science.
Are DB trips for lookup fetches expensive in today's world?
This question of yours expresses a fairly common misunderstanding: that it can be answered sensibly in general. To my experience the only sensible answer is: it depends (on the DB, the application, the whole system, the network etc.).
But let me answer that question differently: yes, it is possible that lots of DB roundtrips can become expensive in a system, especially when the network has a high latency (which is not uncommon when accessing a DB over a wide area network, with lots of intermediate routers, VPN channels, data encryption etc.) However, nothing except you own measurements and observations can tell you if this matters in your system.