Giving you the answer (IE writing the program for you) doesn't really help you learn the language, so here's a pointer in the right direction:

rand^5+1->X
[0102030405]->L1
"ABCDEF"->Str1
3*x-2->Y
.perform Substring operation on array pointed to by Str1.
.Be careful to handle the case where the starting location is greater than 3, as will often happen if X can vary between 1 and 5
For(X,1,20)
End
.no need for ipart in Axe, since Decimals aren't supported.
5->A

Be sure to keep in mind that you can't always just take BASIC code and easily translate it into Axe, particularly with the high level commands like Sub(). Axe might look like BASIC, but it requires a slightly different mindset.


EDIT: I'm going to move this into the Axe discussion board, since it's more likely to get help  and the topic fits in with that board better.

Hi, in the future, try to avoid double posting like this. You can just use the quick modify button in the top right of your post to add additional thoughts.

also use code tags [ code] [/ code] w/o the spaces

For future reference, you don't need to exclude spaces if you write it like this:
[nobbc][code] [/code][/nobbc]  (Which looks like [code] [/code])

sub("ABCDEFGHIJKLMNOPQRSTUVWXYS",3x-2,3) means:
if X=1     "ABC"
if X=2     "DEF"
if X=3     "GHI"
...


which commands are also different?

Data(65,66,67,68,69,70)→Str1A  // Note that 65 is the ASCII value for 'A', 66 for 'B', and so on.
Zeros(4)→Str1
For(A,0,3)
{3*X-3+A+Str1A}→{A+Str1}
End

an easier way:

"ABCDEF"→Str1
Zeros(4) → Str2
Copy(3*X-3+Str1,Str2,3)

I know I'm a bit late to join the party to help out here, but I have a lot more information to offer. What follows is a list of code conversions for all your examples, each accompanied with an exhaustive list of notes. These notes specify how the Axe code works and help outline how it is different from its BASIC counterpart.

If you have any questions left unanswered by this, please ask!

BASIC		Axe		Notes
randInt(1,5)→X		rand^5+1→X		rand returns a pseudo-random integer from 0 to 65535. ^ is the modulus function, which would then limit this number's range to 0 to 4. Adding one increases the range to 1 to 5. Axe 1.0 may have a built-in random integer function with a specifiable range, making the above obsolete.
{1,2,3,4,5}→L₁		Data(1,2,3,4,5)→GDB1		Instead of storing the list of numbers in an OS list variable accessible outside of the program, the data is stored inside of the program. If this is compiled as an application, you will be unable to change the values in this list. Get around this by copying the list to RAM and instead operating on the copy. Each number is only a 1-byte integer instead of a 9-byte float. Get used to only using 1- or 2-byte integers in Axe! Whereas code to access this list in BASIC would be like L₁(3), the Axe code would be {3-1+GDB1}. The -1 is added because data in Axe is zero-indexed. You could make this a list of 2-byte numbers by adding the ʳ modifier to the end of each number. Elements of the list would then be accessed with code like {3*2-2+GDB1}ʳ
sub("ABCDEF",3X-2,3)→Str1		sub(SUB,"ABCDEF",X*3-2,3)→S .Later in the code: Lbl SUB 0→{Copy(r₂+r₁-1,ᴇ8000,r₃)} Return ᴇ8000		Axe has very few built-in string manipulation functions, which is why... You supply your own routine! Lbl SUB is a custom routine for the substring operation. It places the substring starting at byte 0x8000 in memory in a 256-byte section of free RAM not used by anything else in Axe. sub(SUB,[arg1,arg2,...]) calls the routine named SUB, with up to 6 arguments optionally specified. These arguments are then placed in order into the variables r₁-r₆ Note that implicit multiplication (e.g. 3X) is not supported. X*3 is used instead. 3*X would also work, but it much less optimized. The result of the function is not an actual string object, but a pointer to the object. So the value returned is stored to the integer variable S. Get used to strings, OS variables, and more being referenced as pointers to the objects, not the objects themselves!
iPart(X)+abs(Y)→Z		X+abs(Y)→Z		Since values in Axe are 2-byte numbers, floats and decimals along with functions like iPart() are out. Because certain code can result in more optimized assembly, this would be more optimized as abs(Y)+X→Z
For(X,1,20,1) End		For(X,1,20) End		Axe does not support an increment argument with For() loops, so it's a good thing your sample code had an increment of one! Axe 1.0 will hopefully support the optional increment argument with For() loops I don't know if you have no code inside of the For() loop because this was just an example, or if you were using it simply to generate a pause. If you are just aiming to generate a pause, Axe's Pause # command should be very helpful. A Pause 1800 approximately correlates to a one-second pause at 6MHz, or approximately a Pause 4500 at 15MHz. (Unless you use the Full command in your program and it is executing on a non-83+BE, your code is running at 6MHz)
dim(L₁)→A		Data(1,2,3,4,5)→GDB1 []→GDB0 GDB0-GDB1→A		This code assumes L₁ is the list you defined earlier. GDB1 points to the first byte of the list, and GDB0 points to the first byte after the list. So subtracting the two gives you the distance between them, in this case the size of the data. If this were a list of 2-byte numbers, you would need to make sure to divide the distance calculated by 2. To make a dynamic list with a variable number of elements, you would need to put the list in a section of free RAM somewhere and keep track of the number of elements yourself.

Pt-On is the sprite drawing command which takes 3 arguments.  You want Pxl-on to draw pixels.  Also, you seem to be under the impression that it is easy to convert a BASIC program into an Axe one.  This is definitely not the case.  Even though they look the same, they function completely differently.  Luckily, Axe comes with some documentation of how it works written for people already familiar with BASIC, just like you.  You should definitely give it a read, its not that long.  Also, be sure to check the commands.htm file before using any command in Axe just to make sure it does what you think it does.

Oh and by the way, if you press [prgm] right after an error, it will scroll to the error.

EDIT: And also, you'll want a DispGraph at the end to display the buffer on the LCD.  And starting text with -1 doesn't do what you think either, you'll need the "Fix" commands.