Having owned a Mark V, and pretty much being an amp tweakhead, I have a few tips - maybe you already know them, but here I go anyway. The amp itself is certainly capable of an awful lot of great tones (though it's not going to sound exactly like, say, a Plexi or Vox without a pedal designed to do that, etc).
The first thing to think about is that with Mesa's cascading gain in the preamp section, how hard you drive the first preamp tube is going to make a pretty big difference. To a degree, this is true with any amp, and Mesa is but one example, but Mesas cascade so it's even more a thing. First and foremost, a Mesa is all about the effective use of the preamps, if you don't get that happening in a good way, the power section of the amp can't make up for it.
How often do we dial in an amp, and forget about the output level of the guitar and its controls? But the guitar's output level really affects how the V1 preemp tube responds, and that, in turn, also affects how the other preamp tubes respond. There are lots of "in between" sounds to be had merely by working with your guitar's volume control along with the amp's controls. Same with the guitar's tone control. How hard those first tubes are driven by the pickup can change the character of the distortion, as well as the amount of distortion,. Clipping was so named because the distortion "clips off" the high frequency content in the original signal (and of course it can add its own harmonics). Change that high frequency content, of course, and you do get a very different tone. And how you set the EQ will affect how the gain stages sound when they distort, too.
Also, the amp has both passive (the normal bass-mid-treble controls) and active equalization, and they come at different stages of the amp's innards and even work at different frequencies. Thus they affect the tone in different ways, and you can create some fairly complex EQ curves that will really change things up if you're willing to experiment a little. This is a trick well known by engineers using Pultec EQs, that you can boost and cut various frequencies - even the same ones - using the slightly different curves that are inherent in the EQ and get some really nice things going tonally.
I follow the recommendation of folks like David Grissom, who do things the old school way we did back in the 60s when amps didn't have Master volumes or preamp gain controls. If you set the amp up with the guitar volume about halfway up, I mean really get it sounding good, with a light crunch, you'll find that rolling back the guitar volume makes your sound cleaner, and rolling it up increases your gain and color. This is even true on the clean channel, especially if it's set to Tweed (my fave MarkV setting for clean). But the point is you're affecting a lot of things with the guitar volume, and you might be able to capture some of the more elusive tones that you're not quite getting to that way. It also works on the higher gain channels, and moreover, it allows you to turn the power amp section up, to let the power tubes work better and affect the tone.
Players who simply turn the guitar controls up all the way to set up the amp miss all this tone potential, and worse, they have to turn down the power tubes and never get the amp cooking all that well. If they're playing live, they have nowhere to go really when they want to quickly change the tone, because if they want to turn the guitar down, everything is suddenly dull, and they can't go bigger either. They're stuck.
But if you have a nice high frequency balance going with the guitar slightly rolled back, then you can affect things either way simply with the guitar's volume, make it brighter, make it cleaner, increase the gain in small increments, etc.
People often criticize Mesas unfairly by saying that there are only a few "sweet spots" on each channel, but I find that is not really the case if you're willing to use the guitar's controls to their full extent.
I stopped with my blather but, heck, I can't resist a few more comments. Speakers make a very big difference. Mesas can do an awful lot of things, but everything is coming out through that speaker cab, and it's certainly worth experimenting with cabs a little. Also, if you're not using really good cabs (you didn't say if you're running a combo or a head, or which speakers/cab you're using), the amp isn't going to work to its potential. Cabs do sound different from one another, and their design and materials are important parts of the equation, not just which speaker is bolted into place.
Final tip: Get a good sounding buffer for your pedalboard, and use a 10 foot cable from guitar to pedalboard (or amp) instead of a 20 footer. It really does make a difference, especially with a high quality cable.
I hope this helps, and forgive me if you're already doing this! The Mark V is one of my favorite amps, ever. Good luck!
