How to set permissions so that I can read and write to another partition?

What I would do is the following:

Assuming you have both partitions mounted with the names part1 and part2, you will be the only one using them and you want total free control over them, I would do this:

sudo chmod 777 /media/part1 - This would give all permissions (Read, Write, Execute) to you within the part1 partition.

sudo chmod 777 /media/part2 - This would give all permissions (Read, Write, Execute) to you within the part2 partition.

The permissions (in this case 777) are as follow:

7 - Full (Read, Write & Execute)
6 - read and write
5 - read and execute
4 - read only
3 - write and execute
2 - write only
1 - execute only
0 - none

The first 7 (Starting from the Left) is for the owner, the 2nd is for the group where the owner resides. The last 7 is for other groups. Basically like this you can copy anything you want in the partitions and if you ever need to take the HDD out and connect it to another computer with Ubuntu you will not have any problems with permissions. At least in my case it saves me time because I tend to have 1 or 2 hard drives that hold movies, music and similar stuff and I move them around from PC to PC.

Just to add, if you do not know where the partitions are mounted, you can always open Disk Utility and in the information about the hard drive it will tell you where it is mounted. Remember that you need to apply this to the partition AFTER it has been mounted.

Instead of changing all file permissions, like Luis Alvarado suggested, it would be better to change the file owner - thus keeping the executable bit on any binary and script files that previously had it.

So, assuming your partition is mounted as /media/something and your username is johndoe, you can run

sudo chown -R johndoe:johndoe /media/something

to change the owner and owning group of /media/something (and all files and directories it contains, hence the -R for 'recursive') to johndoe.

This way, all files will retain their permissions, but since you will be the owner of /media/something, you will be able to write to it and change any file permissions, in case you ever need to.