I'd like to write a little DOS program (my first one) and I'm a little bit inexperienced.
For the program, I need more than 64 kilobytes of (conventional) memory. How can I get extra memory? Ideally, I'd like to have two extra 64k blocks of memory for the program. Can I just start to write data somewhere into the address space or do I need to request extra memory?
Under DOS, yes, you can just start using another segment of memory. There is an important caution, however!
Have a look at a memory map for the version of DOS that you are using. You want to be sure that you aren't selecting a region of memory that is actually reserved for another purpose. Here is one from Dr. Dobb's Journal:
Address (Hex) Memory Usage
0000:0000 Interupt vector table
0040:0000 ROM BIOS data area
0050:0000 DOS parameter area
0070:0000 IBMBIO.COM / IO.SYS *
mmmm:mmmm BMDOS.COM / MSDOS.SYS *
mmmm:mmmm CONFIG.SYS - specified information
(device drivers and internal buffers
mmmm:mmmm Resident COMMAND.COM
mmmm:mmmm Master environment
mmmm:mmmm Environment block #1
mmmm:mmmm Application program #1
. . . . . .
mmmm.mmmm Environment block #n
mmmm:mmmm Application #n
xxxx:xxxx Transient COMMAND.COM
A000:0000 Video buffers and ROM
FFFF:000F Top of 8086 / 88 address space
The "official" memory allocation mechanism is through memory control blocks (MCB) and the DOS interrupt 0x21 using 0x48 to allocate and 0x49 to free memory. A good discussion of this can be found in this Microsoft support document.
For documentation on the interrupt approach, you might look here.