Early on, try gaining exp and ap by fighting Bandersnatches, just south-east of the Wind Temple. Bandersnatches have 120HP, and give 15 exp and 1 ap each. This doesn't sound like much, but compared to the other monsters available, it's a fair amount. Bandersnatches can be tricky opponents, so remember to save between battles.
Early on in the game (i.e., as soon as possible), learn the skills Secret (from Thief), Barehands (from Monk), and Learning (from Blue Mage).
Secret will aid you while searching through dungeons, while Learning will, hopefully, ensure you don't miss any blue magic. Barehands will allow your character to continue to provide a decent offence while studying White or Black Magic.
Harpies are very tough opponents, mostly because of their high defence and attack; they are almost impossible to defeat the first time you meet them.
However, after receiving the Berserker job, you will be able to defeat them relatively easily. Have every party member as a berserker, and heal between battles, and you should be able to defeat enough harpies to be able to pick up the 2000gil, Speed magic and Elf Cloak hidden underneath Walz castle.
A good place to level in the first world is around Easterly Falls. On the plains you will fight groups of five BlackFlames, which give over 800 GP, over 300 EXP and 3ABP. You may also fight groups of two BlackFlames and one StoneGolem, which give a little less GP, EXP and ABP. Both groups are easily defeated by summoning Ramuh.
In the forests, you will fight either Mythril Dragons (and if you have the control skill you can use their Pep Up skill to replenish your HP) or groups of mini-dragons. Mini-Dragons can be difficult to defeat, but the Aqua Breath Blue Magic works well against them, and you gain 1125 EXP on their defeat.
You'll notice that, at Walz castle, there's one crystal shard you aren't able to reach. This is the mime jobclass, and you can pick it up once you have the submarine.
Head to where the Tower of Water used to be; you'll need to run through the tower in under seven minutes (although you can pick up another breath of air in a treasure chest, taking your time back up to seven). At the bottom, you will face Gogo.
This battle is one a lot of people have trouble with. All you need to do is exactly as he says - mimic his attacks. And he'll be mimicking yours. You don't move, he doesn't move. You'll need to have three minutes or so on the clock, but eventually, Gogo will admit that you are worthy mimes.
You can gain Ramuh any time after getting the Fire Ship. Head to the north-western most dot on the map (you will need to go south and approach it from above or go around the continent, due to the way the land is laid out). This village is Easterly Falls. Wander around in the forest, and you'll be plunged into a boss battle with Ramuh. If you are on level 21, the Goblin Punch blue magic will defeat him with one move. Otherwise try using other strong magics, such as Aqua Breath, Shiva or Ice2. He has 4000HP, and on defeating him, you'll gain the Ramuh item. Use this in the menu and he will be added to your summon list.