I have a basic understanding of tile maps and movement, but I'm having trouble moving beyond that. My goal is to replicate the movement seen in NES games like Dragon Warrior and Final Fantasy. Here's the code I've been experimenting with:

function love.load()
	x = 0
	y = 0
	dx = 0
	dy = 0
	size = 64
	t = 10
	v = size / t
end

function love.draw()
	love.graphics.rectangle("fill", x, y, size, size)
end

function love.update(dt)
	x = x + (dx - x) * (v * dt)
	y = y + (dy - y) * (v * dt)
end

function love.keypressed(key)
	if key == "up" then
		dy = dy - size
	end
	if key == "down" then
		dy = dy + size
	end
	if key == "left" then
		dx = dx - size
	end
	if key == "right" then
		dx = dx + size
	end

	if key == "escape" then
		love.event.push("quit")
	end
end

Currently I can press a button and the block will move a set distance, which is great, but it moves at a variable speed; it slows down as it approaches the end of it's movement.

Am I on the right track at the moment? I feel like I'm missing something in the math department.

you are moving by a set amount of distance, but that distance is recalculated each update and getting smaller and smaller

this means the greatest speed will happen at the start when the distance is largest, and progressively get smaller and smaller

what you are doing is equal to linear interpolation: x = x * (1.0 - w) + new_x * w;

x = x + (dx - x) * (v * dt)
y = y + (dy - y) * (v * dt)

Each of these lines calculates the current distance from "here" (x,y) to "there" (dx,dy). The greater the distance, the bigger the step you make. When approaching (dx,dy), the distance shrinks down to (0,0).

What you probably want (and I'm not familiar with the movement system you mentioned in examples) is to get the direction but not difference from "here" to "there". This means movement towards (dx,dy) but with a delta only influenced by speed v and elapsed time dt. This is done like so:

ddx = dx - x
ddy = dy - y
ldd = ddx * ddx + ddy * ddy
if ldd > 0.00001 then
    ldd = sqrt( ldd )
    ddx /= ldd
    ddy /= ldd
    x = x + ddx * v * dt
    y = y + ddy * v * dt
endif

What I do is store the actual character positions in tiles, and a separate pixel offset used for the visual movement.

Say a character moves to the right, add 1 to their tile X position, and set the pixel offset X to -tilesize. Each frame, add or subtract their movement speed (usually 1 or 2 pixels per frame) to the pixel offset to go toward 0.

For collision detection between sprites, only use the tile position. So as soon as a character starts moving, their old position is empty and new position is full. The gradual movement is purely a visual effect.

When you go to calculate the position to draw their sprite on screen, it will look something like

drawPosition.x = entity.position.x * tilesize + entity.offset.x - camera.x;
drawPosition.y = entity.position.y * tilesize + entity.offset.y - camera.y;

So after doing some more Google'ing I found a Game Maker tutorial about this topic, and a demonstration. I managed to get it working in LOVE, but I'd be interested to hear what people think of that solution in general.