Some thoughts:
It takes several, perhaps even many builds before your wiring looks neat. Just the way it is. Electronic wiring, sort of similar to electric wiring (like a breaker panel) has a peculiar combination of delicate and brutal handling, and it takes time to know what you can slam and what you have to be ginger with.
For shielded wire, you can unravel the shielding using a needle from the cut end. Just start picking it apart, try to bend the wires as little as possible until that moment when you are ready to gather the tiny shield wires together. Just work them loose, be patient. Some people like to unravel around and around the inner conductor, evenly. Some people like to unravel in roughly a straight line.
If you want the resistors, diodes, etc to have really super straight leads, you'll have to straighten them. One way is to roll the leads between two strips of wood. Ideally, you need two strips about 4" total wide, maybe a foot long, with a channel routed out of both pcs, the long way. The component goes into the groove, and you move one piece along the other. It only takes an inch of movement.
If you want the bends in the res leads to look good, they have to be pre-bent before they drop into the board, and the bends have to be made w/a needlenose pliers (or, a lead-bending tool, an inexpensive plastic thing you can buy if you wish. Point being...there can't be any lead manipulation or bending ON the board....it has to be "pre-bent"...elsewhere, and has to include the radius of the bend. If I make this sound easy, to get a consistent look of the leads...it isn't. There is always some darn thing that makes you depart from the ideal you start with---either 3 resistors going to one turret...or one lead is too thick to poke into the hole, or you have too many leads to fit into the hole.
When soldering, a clean tip is your holy grail. Damp sponge wipe after every few connections, damp sponge wipe if the iron has been sitting for a while in the base. If you really want clean looking joints...it's good to tin the wire and tin the turret before you solder to it. Some will criticize this comment. A solder connection is a capillary event, it is a chemical reaction, and it has mechanical properties. If the wire and turret are pre-tinned, then the chemical reaction (overcoming the oxides on the metals) has already occurred. Try also not to handle the leads of the resistors with your fingers. Sure, the flux in the solder can and will and is meant to overcome that, and it will. But if you want the super shiny look....
Another part is the distance from a joint that your insulation is stripped back. I don't know any shortcut other than massive practice.
You mentioned you were going to install the heater wires...most of us would do this first. Unless of course you are going to do "up in the air" like a Fender. You don't have to go medieval twisting them tightly, in fact, I reco against it. Twist, yes, but if they are too tight, those wires will not look very good when you unravel them to solder to the tube base.
None of this, by the way, affects how the final amp works or even looks from the outside. So...you shouldn't drive yourself nuts, but as you are doing, look to improve. Your first few builds will look like crap.