There's a lot of personal logic that will go into skill priority. What are your long-term and short-term goals? How early do you want to accomplish certain tasks? Crafting skills, for example, don't train up without you actually crafting so learning them before you plan to start crafting doesn't help. Cooking is helpful but until you get good cooking tools the quality of your food isn't going to be that good.
Early game skill choice, before you start getting all those esoteric skills, boils down to your choices among two categories - practically-necessary skills and nice-to-have-early skills. Or you can prioritize something early if you want it.
I'm not going to cover skills like equipment and Healing/Meditation which are automatic for every playable class and race - only skills that not everyone starts with. I also won't name all of the places you can learn each skill, but mostly the earlier and easier access places.
Practically Necessary
These are the skills that almost every adventurer should have, and not just have but have as soon as possible. The skills I classify here are because they're used almost all the time by your adventurer - and in being used almost all the time, that means they will train the whole time. This makes them doubly-useful to pick up early on - first, because of the obvious benefits of the skill, but equally important is that training a skill trains the stat associated with it. There's essentials for every stat besides Constitution, Magic and Willpower - all three of which are covered by the automatic skills of Healing, Meditation, and Faith. The less you need to pay attention to while boosting your skills and stats, the more useful they are, so that's why you should pretty much prioritize these as early as you can. In whatever order you fancy.
- Weight Lifting (Strength) - Increases your weight capacity by 2s per rank. It's trained as easily as being in Burdened status, which doesn't imply major penalties. As such it's free Strength training and you can always do with more weight capacity. Get it in Yowyn.
- Lock Picking (Dexterity) - Increases your success rate at opening chests and doors. Doors don't need lockpicks to attempt to open them, so even if you break all your picks you'll always train this up as you try to bust doors open that happen to be in every dungeon. More relevant for the Dexterity. Get it in Vernis or Derphy.
- Sense Quality (Perception) - It lets you identify the quality of items you pick up, and even fully identify things over time. It trains every time you pick stuff up, meaning extremely easy means of getting Perception training. Get it in Yowyn.
- Anatomy (Learning) - Increases the chance that enemies drop corpses or parts - the former for food, the latter for cash. Trains every time you kill an enemy, whether or not they drop, so again, free Learning. Pick it up in Palmia.
- Negotiation (Charisma) - Reduces shop prices, increases sell prices. It trains up every time you buy and sell, and the benefit is always nice. Get it at Palmia.
- Traveling (Speed) - In Elona Plus, Traveling trains your Speed stat, which is great. More relevantly, it also causes you to gain skill potential and training in all of your skills in addition to normal experience - all for doing that massive trekking across the world map that you will always be doing. May even be the first skill you consider getting. Get it in Vernis.
Nice To Have Early
These are skills that, like the above, are trained with extreme regularity by the common adventurer. But they're not nearly as essential, and in the case of some, you can only learn them from Guild Trainers that are rather difficult to get early-on. These can be gotten at any time but the earlier you get them, the nicer.
Unlike the above, I've put these in a rough order based on which are nicest to have earlier. The guild trainer skills are all at the bottom regardless of this, but in reality those are almost the nicest to have early.
- Detection (Perception) - In Elona Plus, Detection is the skill used for the majority of gather spots in dungeons. These will be needed for all crafting skills and so it's important to get a leg up - and their prevalence also means that your Perception score will enjoy a healthy boost. Get it in Vernis.
- Gardening (Learning) - You don't need a farm to make the most of this skill. Not only does it make the gather spots viable especially for Tailoring and Alchemy, but it also works for the Harvest Time quests and speeds up your picking. It also makes those quests train Learning like mad. Great for early money and platinum earning. Get it at Yowyn.
- Magic Device (Magic) - In Elona Plus, they added potion usage (both drinking and throwing) to enhance Magic Device, so getting this skill early helps get a nice boost. Spellcasters and non-casters alike will enjoy the bonus ranks in the skill. Get it at Port Kapul.
- Gene Engineer (Learning) - This is the third Learning skill that doesn't really need its primary benefit to be useful. In Elona Plus, certain gather spots use Gene Engineering as the relevant skill. Taking this skill early not only helps pick up materials for Alchemy, but it also gets you a head start as Gene Engineering is one of those skills that you want really high. Get it at Port Kapul.
- Mining (Constitution) - The ease of breaking walls is a side benefit, the major benefit is for gather spots, which is helpful for Carpentry and Jewelry. It's also a good source for Constitution if you need to pass time. Find it in Vernis.
- Fishing (Perception) - You'll eventually use this to actually fish, but until then, it is for gather spots especially for Alchemy. Get it in Vernis.
- Literacy (Learning) - The book reading isn't useful to everyone - the more relevant point is that it trains by reading scrolls. As you'll be reading scrolls often, it's free Learning experience. Get it in Palmia.
- Swimming (Speed) - Increases Speed when in the Wet status - it trains not just when you're in water, but even when it's raining. Or if you feel like throwing potions at yourself instead of drinking them (which is actually a good idea). As one of the only two means to get Speed trained, great to have a jump on. Get it in Port Kapul.
- Jeweler (Dexterity) - The crafting skills usually don't need to be picked up until you start crafting, but Jewelry is a great one to start off early. With Detection (pick that up already!), you'll easily gather the materials even at the very start, compared to the other craft skills, making yourself on the way to easy riches and utility. Get it in Palmia.
- Performer (Charisma) - If you have an instrument, and plan to use it, you might as well start early. Not only do you need it for anything resembling success and not resembling death in Party Time quests, but it also earns you some nice coin. Get it at Port Kapul.
- Tactics (Strength) - Increases the damage you do in melee combat, increased by fighting anything in melee. Straightforward and desirable for anyone unless you plan to never swing your weapon. Those who don't start with it (mostly melee classes) will need to get to the Fighter Guild Trainer for it.
- Greater Evasion (Perception) - Gives a second dodge chance. Doesn't train necessarily fast but you want it early so that when it matters, you'll have built up a lot of experience. The Mage Guild Trainer has it.
- Eye of Mind (Willpower) - Adds critical hit rate. Extremely wonderful and trains very easily since it's very momentum based and makes itself easier to train. Unfortunately, lest you start with it, you need to pick it up at the Fighter's Guild Trainer.
- Marksman (Perception) - Tactics as applied to non-throwing Ranged attacks. Extremely useful and you'll always use it as long as you're shooting things. Both the Fighter's Guild and the Thieve's Guild have it.