Yeah, gaining positive standings with an Empire Faction will incur both positive and negative derived standings with all other Empire Factions.
For traveling around in high sec as well as getting reduced fees for Market / Broker or Reprocessing materials within NPC stations, the only Faction standings that matter are for the Empires that control high sec systems :
Amarr Empire
Caldari State
Minmatar Republic
Gallente Federation
Khanid Kingdom
Ammatar Mandate
The Interbus
Concord Assembly
Granted you don’t really want to do missions to keep your character neutral or positive with the other high sec Factions but unfortunately if you want to increase standing with a specific Faction, you’ll also have to work on the other main Factions as well.
The easiest quickest way to do that is to take the info listed in the ‘Faction Standing Repair Plan’ and consolidate it into steps pertaining to your specific situation. In other words, since you’re only at -1 with Caldari, you wouldn’t have to work ‘The Plan’ very much to have the standings for all main Factions to be close to equal.