How do I tidily join up lines in illustrator?

Question

I'm going to start by saying I'm not the most experienced designer and I'm sorry if this question seems really silly but I am trying to tidy up a logo and I can't seem to get things to do what I want them to.

As you can see my lines are looking very untidy and I have been playing around with the different tools but haven't managed to get anywhere.

As we zoom in it is clear what a mess this is. I've tried playing around with the anchor points but it just ends up looking worse. I have tried looking on YouTube but had no luck with anything helpful to my situation.

Could someone please assist me so I can (a) learn by it and (b) move on to the next stage?

UPDATE #1

Thank you very much for your nicely structured and detailed answers, I really do appreciate it. However I am still not getting the results I'm looking for. It has worked for you but when I join the two anchor points I get the following:

What I did was hold down shift while having the direct selection tool selected (white arrow) and then went to object, paths, join and then this happens to the lines i try and join.

I gather I am doing something wrong, but can't figure out what. I also tried aligning the x/y points so they are a perfect reflection of each other but it does the same.

UPDATE #2

I have since managed to join some of my lines and have tidied it all up a little but for some reason other lines don't want to join and i get this message.

Is this something to do with my layers? I don't understand it if I'm honest. What exactly is a text path?

Answer 1

Good to see that you're trying things out the command you're looking for is called Join - the keyboard shortcut is Command-J.

Two line segments, anchors touching, lines unjoined, with strokes of a fair width so you can easily see them not being unified or joined at this point.

You can see now the anchors and line definitions: the anchors are indeed co-incident (that is, they directly coincide at the same XY spot in the document):

Object>Path>Join command pulled out in palettes for clarity:

Lines correctly joined:

Closeup to show line joint:

Note that you can either Join with the entire path selected with the black selection arrow, or with the specific anchor points selected with the direct selection white arrow, but not with the line segments selected with the direct selection white arrow.

Post-scripts based on OP secondary issues:

To JOIN the paths, it's best practise to have the anchor points be LITERALLY co-incident - that is, you use snapping to make dead certain the two anchors are in EXACTLY the same place: this way you get predictable results.

The popup the OP has seen is typically what comes up when either not both individual anchor points are selected with direct select or the line segments have been selected with direct select - see my above note about this condition.

Answer 2

Joining lines isn't the only missing thing, the vertical spacing of the parts is obviously wanted to be even.

Select a shape, goto Object > Transform > Move, type the needed vertical shift, click Copy. You get a shifted copy. Create more copies by pressing Ctrl+D.

ADD due a comment by usr2564301: Distribute can also used to create even spacing. The shapes can be different when applying distribute.