You should keep the mortar a consistent 3/8 inches thick between blocks. This is true for the side joints as well as the joints between rows. However, you need to put more than twice this much down before laying the blocks in place. This is because the blocks will compress the mortar that they are laid on, and you need to account for this if you do not want to end up with joints that are too thin.
If you are building a wall that is meant to withstand great weights or enormous lateral pressures, such as retaining and foundation walls, then you will need extra-strong mortars such as type S or type M. Otherwise. a general-purpose type N mortar will do the trick. However, if you are going for a historic look to your construction, and it does not have to withstand much pressure, then use an archaic lime mortar, which contains lime, sand and water but no cement. It is much weaker than modern mortars but does have an antique look that may be more important to you.
When your building blocks are pressed into the mortar on the row below them, you are supposed to trowel off the excess that is squeezed out from between them. However, there are many ways you can do this to produce different looks. A full joint has the mortar come up evenly to the edges of the blocks. A raked joint has a shallow groove scraped into the face of the joint so that the edges of the blocks stand out. This is the method most commonly used for brick walls. Overgrout joints have the excess mortar smeared over the faces of the blocks around the edges. This makes the blocks look smaller. Bead joints leave a mound of mortar squeezed out of the joint, projecting slightly beyond the faces of the blocks. Weep joints leave the excess mortar to ooze down the faces of the blocks below them for a rustic look.
You can build a wall out of cheap and functional concrete blocks and still get the look of a natural stone wall with the aid of mortar. Once the concrete block wall is finished, cover it in mortar and attach a narrow facade of thin natural stones. This will leave the wall with the appearance of having been built entirely out of natural stone.