I'm not going to go into any super details on where to go but I would like to share with you how to solo/duo post 1.19 patch. Without taking any delays here is the information:
1) Stop fighting monsters that are above your current level. Stats are now working properly--yes they were grossly malfunctioned prior to this patch--so you have to make a conscious choice on what to fight.
2) If you are soloing, stick to monsters that are generally squishy (low HP, low def) I have come to realize that pteroc type monsters have become considerably easier to solo now than before--just to note--because it seems like they have less HP now and its much easier to avoid the blackflip move (you just move to their side and it lowers the acc of the move considerably if you didn't know) since almost all the battle lag has disappeared. You will net around 140-250 exp a monster from the level ranges of 20-30 (this is based on my experience)
3) If you are duoing with someone--first I would suggest and healer type and a melee type class--stick to monsters that are of equal level to the highest player or 1-3 levels above. Again pterocs in my case have been excellent targets for my level.
4) Wear appropriate gear! Look at your gear, and look at the stats, are you getting good benefits from wearing it? That Onion Knight gear sure does look spiffy but I doubt it will protect you from a monsters TP attack. So make conscious choices based on what you are fighting.
5) Materia, materia, materia! Yes I know it was only introduced the other day so you might not have any created yet, but as soon as you can get it...GET IT! It helps so much at any level, and don't be hesitant to equip a grade 1 materia on a level 20 piece of gear...more than likely when you get to your next gear set you'll be turning said lv20 piece of gear into materia anyway! So use it up!
6) Exp chains and linking, so you want exp huh? Its quite elementary then dear sir, and here is how you can do it. First pick off a stray monster and pull him away from the group (every class has a ranged attack so don't you dare say you can't do this) Whittle it down while keeping it away from other monsters then before you kill it pull it closer to the other monsters so another can "link" with them. Voilà! Rinse and repeat your method and you will have link and chain experience galore!
7) Rest when needed, your HP/MP doesn't regen as fast as it used to when you run around prior to this update so tread carefully and stop to rest when needed! I don't think any further explanation is required.
8) Don't underestimate the value of the dungeons (all 3 of the them!) They are absolutely great places to gain exp in solo/duo if you are the appropriate levels. (in fact I gained 3 levels in three toto-rak runs yesterday, sure we were not able to get a full clear on the dungeon but we were looking for exp anyway; that's levels 27-30 duoing)
9) Side Quests and Guild Leves! My oh my how easy those side quests are, some don't even require you to fight anything! Free EXP? Yes, please! Also we get gil and/or various items and some nifty lore to boot, why would you not be doing these? Also mind you Guild Leves are excellent ways to gain exp, even better I'd argue post-1.19, just make sure you follow the guidelines above when picking the monsters' levels, and you should do fine.
10) Finally, make sure you have some cross-class abilities! You are never going to get anywhere if you think you can solo without: Bloodbath/Second Wind/Rampart/Cure/Sacrifice/Protect/Shell (Pro/shell might not be required as I made it to 30 without having it just fine; but you have to have ways to heal yourself!) Not to mention that all those abilities save for the last two are learned before level 10 you should have no problem getting them.
Okay so there are 10 things to consider when soloing, I hope this helps the masses that are lost in the dark about why they can no longer solo, because that is false, I am soloing just fine with little down time; and you can too!