Well, what you've said there is very vague. Any weapon, with enough skill, will get you through any level.

What you need to choose, is a role. From you weapon choice it seems you want to be a close-mid range DPS attacker.
So I'll go from that and try to offer some advice.

First off, if you want to solo effectively, memorize this chart. Then memorize every monster in each category so you know which ones are what and then, through playing the game, where they appear.

Now I'll give you some insight on your weapon choices:

The Dread Venom Striker is a great sword, it deals good damage, it's normal typed so when it poisons an enemy (which is fairly likely), it gets an outright boost in damage, too. It's a good DPS weapon, poison in an excellent status and it has good synergy with normal damage for it's defense debuff. It's charge attack, when done right, can also hit quite a large number of monsters at once. I've hit 5 jellies once ad managed to get to lumbers with it another time.

The Blitz Needle is another good weapon. Since it is effective against two of the best dodgers in the game, Wolvers and Devilites, it is more than worth your time acquiring since a few blasts from close range will take them out now problem. It's low damage per shot means it's not a weapon you can use against everything, since the penalty for attacking monsters that are strongly resistant to your attacks will bring your damage down quite a bit.

The Volcanic Pepperbox, this is the Dread Venom Striker of Guns, really. I wouldn't recommend it in a set if you were going to use both of the above weapons. In fact, unless you're using all guns (or aren't using a Dread Venom Striker) there's not really a reason to have this gun at all. That's not to say it's a bad weapon, but if you have close range weapons at your disposal you don't need to keep away monsters, just let them come to you and wipe them out with a Dread Venom Striker charge attack.

Radiant Sun Shards is an excellent bomb. It's Elemental/Piercing Type Damage makes it quite deadly to a wide variety of monsters, but it's little brother, Ionized Salt Bomb is a little better in this regard, because it's hybrid pure damage (normal/piercing) means it is more well rounded, plus it deals the shock status. The Sun Shards are an excellent Trojan Killer though, they also hurt Undead a little more than the Salt Bomb, too.

Venom Veiler requires an entirely different style of play to use effectively, since monsters that come out of that poison cloud will lose their poison status fairly quickly, you need to keep crowds controlled while under constant damage to use effectively. Something that is hard to do without Crowd Control specific weapons.

I would recommend these weapons for you:

1. Dread Venom Striker: Good effect, good synergy with status, high damage.
2. Gran Faust: Good effect, good damage typing (normal/shadow), very high damage and knockback. Also unique in the sense it is the only weapon to inflict Curse.
3. Magma Driver: Inflicts fire (flat DPS), bullets rare get dodged, moderate speed. Also helps you break some puzzles.
4. Callahan/Blitz Needle: Either one will help you deal with the manic dodging of Wolvers and Devilites. But the Callahan has a really long range charge attack that will allow you land a cheap hit on most monsters. It also stuns, so you can close the distance and wail on them with your Dread Venom Striker or Gran Faust.

But you can make up your own mind by looking at the wiki some more.

As for armour, the choice shouldn't be to cover weakness, but accentuate strengths. Once you get good enough, damage will be a thing of the past. Being defensive won't hurt, but always keep in mind how an armour helps you before you worry about how it might hurt you.
Some penalties are just worth it.