There is no best overall ship. It all depends what you want to be able to do.
You want to live forever? Fleet Galaxy-R. You'll live forever, but you won't ever kill anything either.
You want to DAKKA DAKKA DAKKA... and MOAR DAKKA? Fleet Defiant-R/Fleet Prometheus-R/Jem'hadar Attack ship. All damage, no brains. Point and click. Easy mode.
You want to screw with an opponent's ship? I wish you luck. Science is so weak now it's terribly easy to counter and only really has a marginal effect on the outcome of a battle (except sub-nuke... which is just cruel... but it's a captain skill, not a ship skill). Your best bet here would be a drain build. Since those still work nicely XD.
I love this question and I get it A LOT from new players. The answers that say "depends on your profession and play style" really are the BEST and truest answer that you can get...however, I will attempt to point you in the right direction...
Now, remember, this is simply a pointing to ship and basic style...there is A LOT to ship building (ie your boffs, and 5 assigned space doffs, the weapon you choose, consoles come into play as well but it varies on quality and what you've stacked, and of course the space sets claimed throughout the rep system and how you've set up your skill tree)
Now, everyone typically makes a Tactical captain first. Probably because in TNG, DS9, and Voyager, the captains were in red, tac is also probably the easiest, and by easiest I mean most basic profession as it is all about destroying your opponent. But whatever the reason, I will keep this geared towards tac because my hunch tells me that if you're reading this, you're new, and have created a tactical captain.
Now, I'll be honest. Prior to the nerfing of Going Down Fighting, the bug ship (aka Jem'hadar attack ship) doing alpha strikes was probably the most powerful, highest dps, best survivability, nastiest little ship in the game. This ship is also the most sought after, hardest to get, and most expensive ship in the game. Don't believe that? Go look on the exchange...currently there are 6 and they all cost over 400 million EC. That translates to approximately $400 US.
So lets be real, chances are you aren't going to get that anytime soon. It's not available in the C-store, and the random drop was ended in March.
So my next suggestion would be the Risian Corvette. This ship is a bind on pick up ship currently (and only) available during the Risa Summer Event introduced in season 7 June 2013.
Now if you've missed these two ships...my suggestion at this point is to stay with escorts...eventually you will get into a fleet, or find a fleet selling temp access to their shipyard. Any Tier 3 shipyard or higher will have the option to purchase the Fleet Tactical Escort Retrofit. That's as close to a big ship as you'll find.
Remember, as tac, your life is all about dps. Shoot shoot and more shoot. But the escorts give you the speed and maneuverability to stay alive...basic rule, if you can out run and and maneuver weapons fire, your enemies will "miss" you.
Now we get into Engineers. These are your space tanks. They're the guys who hold agro and take the pounding. For these guys, obviously you're gonna want to stick with the larger ships like cruisers. Think of them as big old world war 2 battleships...they are designed to take a beating but properly set up, you can do some shield stripping or mild/moderate dps.
And last but not least, we have science. Probably the more difficult but most versatile and rewarding profession in game, your science toons are healers, debuffers, and your jack of all trades characters. Depending on how you've set up your skill tree, you can have a science officer do some debuffing and moderate dps...or be a healer or, in some cases, tank well. Right now, I'm at a toss up between the Vesta and Wells class Temporal Science ship. Basically it comes down to what I want to look at on any given day. Vestas are great (especially getting the bundle) for their aux cannons. This lets you dump everything into aux which is Sci's best friend. This is how my Vesta is set up. But the temporal ship won't use cannons, so you're stuck with beams, or you can set this up to tank and be an all torpedo ship.
Remember, you need to figure out what you're most comfortable doing and go from there.
Once you've chosen a ship, choose ONE energy type. In game we call the noobs with two or more different energy types "Rainbow nooblets" or some variation of this because, well obviously you have a rainbow of colors flowing out of your ship, and are completely useless to any team you're on, set up like that.
Hope this helps and any further questions, look for my fleet Valor Empire, any one of us will be glad to help you in greater detail.
I tell the new players in my fleet the cheapest and fastest way to get into Elite STF's and and be effective is to do the Dominion story arc and get the Jem space set. Then you can replay to get the Dominion Polaron weps. The boost to Polaron damage from the full set is nice and XII blue Polaron Tac consoles are cheap on the exchange. You can get 4 for about 1.5 mil EC. So now your ship is nice and purple and you are doing decent damage. If your in a cruiser you can go all Beams and Torps and in an escourt 4 DCHS and 3 Turrets. Now you can do STF's and work on doing the Omege reputation and get the Borg gear. A full XII BOrg set with the Omega Torp, cutting beam and console and full space set is great on an Escourt. This is advice for the Free 2 Play player who is not going to spend $$$ to get good gear or ships. So grab a Mirror assult cruiser or Mirror Patrol Escort off of the exchange for 30-50k EC and then do the Dominion story arc "of Bajor" Get the Full Jem Haddar set and replay to get the Dominion Polaron Weapons. Do "Gorn minefield and some Foundry farming missions for loot to sell for EC. Once you have 3 or 4 million EC you can get XII blue Tac consoles and some XII Neutronium armor and perhaps a XI blue field generator and your good to go.
Easy answer for this question but the answer creates all new questions. The answer is Any lockbox ship, any lobi ship, and T5 shipyard fleet ship, and any T5 zen ship is the best. How they all are the best depends on the player themselves.
Really OP you should learn to decide what is best for you and not what other people think is best. life is much better when you stop caring about what everyone else has or uses especialy when the question is of a subjective answer.
1. What is your goal? Is it it PVE? (Fleet Alerts, STFs, Dilithium farming?) Or do you want to shame your friends at PVP?
2. If it is PVE, then what class are you? (Sci, Engineer, Tac,)
3. Do you want to tank? Run Sci Skills or DPS.
4. What is your skill level. You can put a great player in any ship and win or a lousy player and struggle. If you are a Sci, then there is a even greater premium on skill.
5. Science: The vesta is probably the best, but you can have great success with the Nebula and others. The key is to have strong aux power levels and know what skills to use.
6. Engineering: Are you a tank or Dps? Guides the trade off between tac consoles and engineering consoles.
7. Tac: It is all about DPS and high weapons power. You need an escort with 5 tac consoles. Pretty hard to beat the tac kumari at this point. Takes some time to learn how to use it. Also, player skill on the minimum tanking skills is important as well. But you should be able to so a scary amount of dos with this ship. Also, you are a tac you are supposed to be a glass cannon.