My first few times in T3 were a bit shocking as well. Everyone is going to offer opinions, so here are mine:

Plan ahead--look at the gate map and see what levels you are likely to be facing. At the very least, take note of the tier theme.

Have varied options for armor and weapons. Right now your Ash Tail set and rock jelly shield only give you defense against normal and piercing damage. Unless you're just fighting on non-elemental beast or jelly levels, this means you're defenseless against anything else. This includes all elemental and shadow attacks--and T3 enemies do quite a lot of elemental/non-normal damage attacks. Unless you're prepared against this, you won't last long. It's good to have a mix of armors, helms, and shields to choose from depending on the levels (see "plan ahead", above). Try to get gear that also gives resistance to status effects. This goes for weapons as well--the Levi is fine, but won't give you bonus damage to anything. Likewise, the Antigua is pure piercing, and useless against Jellies/Constructs. Different weapons work better on different enemies, and you need to optimize for what you'll be facing.

Somewhat related, don't be afraid to mix equipment sets to give yourself a wider range of defenses. But, again, make sure your mix is appropriate for whatever strata you're on.

Know enemy behavior. Enemies behave differently than in T1 and T2. They are more aggressive and use their special attacks more frequently, and do more elemental/piercing/shadow damage. Other enemies, like gremlins, are quite a bit different. Learn how they work and how to fight them.

I hope that helps a bit.