There is no standard solution, but there are some guidelines. The rest is, frankly, art.
So you require six colors that:
Represent a scale from low to high. This implies the colors should fit on some sort of single dimension themselves. It could be a scale of lightness, saturation, or hue (color spectrum), or a combination of these. Generally, a hue scale only works well if it uses part of the spectrum, not the full red-to-purple range (despite their popularity in weather maps).
Are distinct in bright sunlight. This implies you use highest saturations you can to minimize “wash-out,” and you pick colors from all across the spectrum to maximize their color difference. To numerically compare the distinctiveness of your candidate colors, covert RGB values to Luv and calculate their Euclidian distance.
Work well with users with color deficiencies. This implies you include lightness coding at least in combination with some other dimension. However, with five colors that can’t be all you use since users will have trouble identifying (in isolation on your map) more than three or four different levels of gray. When including hue, it’s better to distinguish colors with blues and yellows than greens and reds since red-green color vision deficiency is far more common than blue-yellow.
Avoid reds and purples. And blues too In addition to using red and purple for other codes, you appear to be using blue for rivers, so blue shades are out. Well, there goes half the spectrum.
Be consistent with user expectations and associations. That is, the colors should be consistent with their meanings. In your case, coding the quality of tracks, we need each successive color to suggest a less robust, softer, poorer quality, and more natural (i.e., not modified by humans) surface given your map backgrounds.
Normally, you’d also be concerned with maintaining a minimal light-dark contrast with all possible backgrounds. Fortunately, it looks like you outline the tracks with thin dark lines so you get good contrast almost regardless of the track color and background.
Nonetheless, you see you’re under some pretty strict requirements that are somewhat mutually exclusive. Trying to balance them all the best you can is the art part. Here’s one way to do it:
Starting with Requirement #1 and #3, lets go from dark to light through the grades. Given you’ve light backgrounds, darker tracks will contrast better and thus appear more robust, so let’s make Grade 1 darkest and Grade 5 lightest, thus fitting with Requirement #5. Using our full range of brightness you have L-values (in Luv color space, not HSL) of 0, 25, 50, 75, and 100. That makes Grade 1 black (0 0 0) and Grade 5 white (255 255 255), so two colors down, three to go.
Now for hues. For Requirement #2, we’ll work our way around the color wheel, picking three colors as far apart radially as we can. However, we’ll only use the Brown/Orange - Yellow/Olive - Green portion of the color wheel since Blue-Purple-Red is out due to Requirement #4. For Requirement #5, I’ll guess that brown (suggesting gravel or dirt) will be associated with higher quality tracks than green (suggesting a grassy or traceless track), so let’s try brown – olive – light green for Grades 2 through 3.
For the exact colors, we’ll max out the saturation for Requirement #2. To give an added boost for Requirement #3, we’ll make the “green” slightly bluish green. We’ll test them in a color-deficiency simulator to see how they do.
Here’s what I came up with as a starting point for you:
Grade 1: 0 0 0
Grade 2: 98 49 0
Grade 3: 105 127 0
Grade 4: 0 210 140
Grade 5: 255 255 255
Which look like this:
Brightness and color distances (about 25 and about 60 respectively in Luv space between adjacent pairs), aren’t great, but about as best as you can do with 5 colors and only half the color wheel.
Putting it though a color vision deficiency simulator shows it’s not a bad start, but could use some tweaking.
You probably can do somewhat better by calculating the exact magenta and blue (and red?) Luv's you plan to use, and getting the 5 colors far from them specifically around the color wheel (at least 50, and ideally closer to 100), rather than banning the entire half of the spectrum.
For more on using color codes, see my Breaking the Color Code post, which also includes details on computing Luv values from RGB.
You stated in your comments that color-coding is your only option. For the benefit of other readers, this example should make it clear how hard it is to get good color codes. Ideally, you’d use some other graphic code for this. Your link to Track Types includes standard symbology using frequency coding (which you can’t use, but others could). Weight coding (line thickness) would also be an obvious candidate, maybe combined with frequency coding. Either or both would make coding that better fits all your requirements.