There appear to be two main ways in which time travel is shown in SF. One is of an unchangeable time stream and the other involves the ability to alter events in the past.
The first of these is exemplified in Heinlein's stories, "All You Zombies", "By His Bootstraps", "The Door Into Summer", "Time Enough For Love" and "Number of The Beast".
In three of these stories, the protagonist goes back in time, has direct encounters with himself, but the sequence of events at each meeting is unalterable, simply seen from a different POV each time an encounter takes place. In "Door Into Summer", Dan Davis never encounters himself face-to-face, but does see himself drive up to Miles house, whereupon he takes his car and uses it later to carry away Flexible Frank. In "TEFL", Lazarus Long, arriving back in 1916, speculates on the possibility that time could be changed, but he never attempted to try and the earlier events dealing with the recovery of Libby's body appear to support the notion that time was not changed. "NOTB" actually has multiple parallel universes along three time dimensions, but it is clear that as far as the time traveling goes, they are traveling in a single dimension during the rescue of Maureen. There is also no 'face-to-face' between the various versions of the characters directly, but they are definitely aware that they are present on several occasions at the same time.
The second type of time travel is exemplified by "Cat Who Walked Through Walls" and "Farnham's Freehold". In both of these stories, time travel is used to change events in the past. "Cat..", a sequel to "NOTB", does allow for changing time lines, The main task of the Circle of Ouroboros is precisely that and for that reason, they wanted to restore the sentient computer Mycroft Holmes from "The Moon Is A Harsh Mistress" and use him to better predict the outcomes of such meddling. Actually, it is not clear in "Farnham's Freehold" what caused the changes, since the changes were already in place when Hugh and Barbara arrive back in the present.
For the unalterable time travel scenario, there is only the single linear time dimension.
The time changeable version can be considered in three separate ways.
1) No real time change at all, but simply the protagonist moves to an 'alternate' dimension where he becomes part of the fabric of that dimension and it moves on, changed from what he remembers. H.Beam Piper, Keith Laumer and others have written about such alternate dimensions but without the notion of traveling forward or backward in time, just laterally. This may be the explanation for what happened to Hugh and Barbara in "FF".
2) There is only the single dimension, but it somehow is altered from what it was. This notion is beset with the possibility of having many paradoxes. Ward Moore's "Bring the Jubilee" seems to be of this type.
3) A 'fork' in time is created whereby the original line remains, but a new altered line is created. Richard Meredith used that in his "Timeliner" series.
Most writers usually have someone in the altered present being somehow aware of the change having taken place, as in Larry Niven's "The Return of William Proxmire" in which, Proxmire trying to destroy the influence that Heinlein had on the space program, saves him from tuberculosis and having him wind up as an Admiral and the staunchest supporter of space travel in the new reality. In some cases, such as Poul Anderson's "Time Patrol" series, the traveler himself or other travelers 'downstream' of the change would know that the changes have taken place and would see them when they returned to the present, but no one else in the present would be aware of them having taken place. In the Moore novel mentioned above, the traveler never returns, but lives out his life in the altered time line, and subsequently after his death, evidence of the event is found in the form of his specially designed watch and diary. Interestingly, the resulting time line is our own whereas he started out in a time line in which the South won the War Between The States.
In Hogan's "Thrice In Time", there was no time travel itself, but the characters could send messages backwards in time and the messages themselves could cause literally everything involved in the change, including all memories, to be totally 'reset', so that no one is ever aware from their own experience of any change having taken place. The only way that they realized that this was happening was that they received messages predicting events which subsequently never too place or giving them warning of events to prevent. He also used the interesting twist that such 'resets' occurred naturally all of the time so the time line was continually being 'reset' on small scales without anyone ever being aware of it, and that the process of sending the message actually caused events other than those directly caused by the recipients of the message to change.
Heinlein took the notion of time travel further than anyone else I know when he postulated the three different time dimensions in "Number Of The Beast" rather than the customary single one, but I am not going to get into that one at this time.
The time unalterable variation has absolutely no paradoxes, but does give rise to questions about 'free will' or 'predestination'. The changeable time version has problems with paradoxes, but minimizes or possibly eliminates the 'free will' questions.
Any time travel story, whether with changeable time or unalterable time can be confusing, but none of them top, IMHO, "All You Zombies".
Below is a diagram which I have drawn to try to make the events in the story clear. The key to avoiding confusion is to start at the right point, in this case, with the birth of the baby at point A. Assume, for the purposes of discussion, that the POV of the character begins at that point in time. Don't worry about 'where did it all start'. Keep your focus at all times, (no pun intended), on the POV of the individual as she/he goes through time and the various loops.
(Note: this might make it appear that there is a 2-dimensional time continuum, but that is not the case. The various lines are simply separated for demonstration purposes.)
A timeline for 'By His Bootstraps' has now been included