I'm running a Fender MIM Strat and a ESP LTD Viper 10 into a Laney LX65R, no pedals or anything, and I'm trying to get a distorted tone with that clean aspect still. This question has probably been asked a fair amount of times but hoping someone here can dumb it down as honestly I'm still a bit of a noob as I've only been playing a year. Appreciate any advice!
How can you have distorted yet clean tone? They are polar opposites.
My Gear:
Ibanez Jet King 2
Ibanez GRG 7221
OLP John Petrucci
Epiphone Les Paul Custom Pro
Squier Stratocaster (modified)
Harley Benton CLD-41S (Acoustic)

Peavey Vypyr 30.

Boss CH-1 Super Chorus
Boss DD-3 Digital Delay
Boss FRV-1 '63 Fender Reverb
I'm guessing here, but I think you are referring to a cranked tube amp sound.

Closest you can get to that is a good overdrive pedal and don't push the gain too hard. Many Master Volume style amps can also do the same thing, set the gain or Pre setting to the gain level you want, I usually use about 7, and the Master volume tot he volume level you need.

This is what I use my Marshall Bluesbreaker Overdrive for, mostly clean but a little breakup. At the volume levels we play, my amp is still totally clean, even the 15 watt Pro Jr, and I want a little edge sometimes, that's what the overdrive is for. If I want a little more, I go to the distortion pedal.
Hmmm...I wonder what this button does...
perhaps an example or two of what you are shooting for would help. sounds like you want a distorted tone with great note clarity to me. still are we talking yngwie or SRV or something else. I play strats as my main guitar and go for a distorted tone that retains note clarity (more in line with yngwie or early 80 gary moore). for this I use a somewhat distorted amp with a TS style overdrive used as a boost.
I'm a clean plus pedals player, and if I understand you correctly, I would be looking at something like a Digitech Bad Monkey set for a subtle amount of grit. The OD is really only evident when I play chords. For an overall grittier tone, but not turning into sonic mush, the Boss SD-1, again set for subtle OD (about 9 o'clock with my pickups) works well for me.
When I hear "distorted clean sound" I think of a tube amp on the verge of breakup. Where light picking creates a light clean sound, while beating the strings creates enough juice to get those tubes to start distorting.

I am not too familiar with Laney amps, but I believe yours is a solid state. The distortion created by clipping a solid state circuit is different than an overloaded tube. So the sound you are looking for may no be able to be found with your amp.

Find you a little 5 watt tube amp and crank that baby up. I have a feeling its the tone your looking for
The Mad Professor 1 pedal does a good job of putting just a tad of overdrive into a clean tone while retaining note clarity, it can do some dirty overdrive tones as well.
"A well-wound coil is a well-wound coil regardless if it's wound with professional equipment, or if somebody's great-grandmother winds it to an old French recipe with Napoleon's modified coffee grinder and chops off the wire after a mile with an antique guillotine!"
- Bill Lawrence

Come and be with me
Live my twisted dream
Pro devoted pledge
Time for primal concrete sledge

If you are looking for a slightly dirty clean tone I would suggest an EHX Soul Food or another Klon clone type pedal, very nice sounding for low gain adding just a bit of hair to your tone and retains clarity.