With two exceptions, there are no synergies to leverage and few shortcuts that can be taken, so the answer is almost “sorry, it just takes time.”
Combat increases by fighting monsters, but that just takes time. Grind that by going through your favourite few easy levels of the Mine (which will vary according to a player's skill and weapon preferences) repeatedly, leaving and then coming back with the Elevator. Foraging increases by collecting forage items and tree (not stump or stick) wood, but forage items and trees take time to grow back so there's no rushing that. Cultivate a tree farm for wood to get reliable gathering opportunities. Mining is increased by breaking rocks, so break lots of rocks while fighting those monsters.
But there are a couple of things you can do that have significant efficiency gains:
1. Turnip Empire
Farming increases per crop harvested regardless of its type. The way to increase farming the fastest is therefore to ensure that you're growing crops that allow you the most harvest actions per season.
This means you become a Parsnip farmer in the spring (6 harvests per plant/square). In the summer you grow Hops (16 harvests per plant), and in the fall you grow Bok Choy (6 harvests per plant/square). In the fall Corn can give the same efficiency as Bock Choy, but this requires planting in the summer and Hops beats Corn in the summer by miles.
You can increase harvests per season further with liberal application of Speed-Gro fertiliser if you have the coin or crafting ingredients for it.
2. Crabbing Empire
The fastest way to gain Fishing XP while still having time to do other things in the day is by using Crab Pots. Though they are only 5xp daily each, they are very fast to harvest from and you can have many of them. Fishing for fish has a better XP payoff for time spent, but when you're trying to get other things done in the day then Crab Pots are the way to go because they leave time for doing other things.
You can speed all of this up by using Warp Totems, a horse, or unlocked minecarts, of course.
Most of the things that improve skill gain efficiency are not cost-efficient, so bear in mind that by aiming to be the most well-rounded character as fast as possible, the trade-off is inefficient wealth growth.
If wealth is considered, the answer returns to “sorry, it just takes time.” If wealth also matters, then just make a cuppa and enjoy country life at its own pace. It doesn't actually take to long to max out all five skills by just playing without paying special attention to them.