Ships & Modules

Drake and then anything else you want.

HM or HAM these days?

That's up to you.

Obviously HAM does more DPS, but HML is better if you're lazy

Been banging this drum a lot it seem lately, but if you want interesting (over the dull Drake - your words);

Caracal - Cerb - Both very good and a Rapid Light Caracal will breeze l3's and then move on to L4 easily enough
Navy Scythe with Rapid Lights is very fast with an AB and has a small sig - lots of fun to fly (can do 3/10 & 4/10 DED's fine too)
Navy Osprey - with HML's is a lot of fun, also very fast and great range for less isk than the Cerb
Sacrilege - with HML's, an MWD and Cap Rechargers is just funny, as you can run about at 80k permarunning the MWD and still have cap to run a repper (but you don't need it cause your are 80k away).

Obviously, I prefer cruiser size ships for their speed and sig radius and the fact that many pack a pretty good punch now.

There is no such thing as a fast missile boat. Even if you have missiles that are as fast as rockets, they are gonna be such close range that you spend half your time just getting in range. If you want to run missions fast, you need to stop using missiles. When it comes to Battleships with missiles vs turrets, it's even worse.

If the OP is doing Lvl 3s, then a HM Tengu, NightHawk or Navy Drake are all very good choices. If you don't have the skills, then a normal Drake will be fine.

I'm using a cerb for level 3's because it doesn't have the DPS to do level 4's and my Tengu's in null.

Tengu > nighthawk > ceberus > navy drake > drake > caracal

If you want to get stupid you can save fits for HM's or HAM's and pick depending on mission. Some missions are quicker if you can reach out to 90 km.

I would still say the drake.

The navy drake isn't worth it because it doesn't have enough PG/CPU to fit 8 T2 launchers and a T2 shield tank. The dev's said this is intentional because when they streamed the new Odyssey features on twitch they said that the navy battlecruisers are bonused for taking out small targets like frigates and destroyers. They aren't meant to be like their T1 counterparts "Tank and Spank" mentality.

As for "HM or HAM". I would say its up to you. You almost need to put on an afterburner or microwarpdrive to get in range with HAMs, but if you have your missile support skills to IV then with HMs you can almost always just slowboat to the furthest enemy in the pocket while picking off everything else along the way. Is this the most efficient way to do it? Probably not but it is the laziest way to do it, and I am the epitome of laziness.

Just get a heavy missile drake and grind it out.

It doesn't take very long to jump to L4's.

Two things can speed you up:

Have someone who can already run L4's get the missions and grind up to L4's with them.
(( Corp with L4 standings could jump this step, although odds are slim you are in or will be joining one ))

Look for a story line agent that is part of that corp to be close by so they story lines will be ran for the same corp.
This option is not always possible but do check to see that you are at the best L3 agent you can run with for your goal.