Hi Rjblanchard -
<<"I'd like to book it before we get there, which will be in late June. Thanks in advance!">>
Many of the special panoramic trains listed by Alpenrose require seat reservations, for which there is a supplemental charge, in addition to the cost of the ticket required to ride the train. In some cases, that requirement might only be for special cars with panoramic windows, as on parts of the Golden Pass Route. Often regular ( "non-Panoramic") trains run the same routes.
However, in general, it is not necessary to reserve seats for travel on "regular" trains throughout Switzerland. In fact, on regional trains, it is not possible. It is possible on the faster Intercity trains.
Trains that are parts of international trans-European routes will have reservations available, and will require them for the parts outside Switzerland. For example, if you were to travel to Milano Centrale from Bern via the Simplon Tunnel, you would not need reservations until you reached Domodossola on the Italian side of the tunnel. And, you would require a separate ticket for the Italian part.
Since Basel is a border crossing point for trains from other European countries, even if you get on in Basel, you may find that many seats are reserved on some trains. Mostly, those are trains to Zürich. You will go via Bern if you go to the Jungfrau Region. By the way, Arno's recommendation is a very good one, and offers flexibility.
Arno has listed a variety of possibilities in his link above for a one-day journey throughout the Jungfrau Region. None of the trains or cableways that he describes require reservations, and almost certainly do not offer them. (He does not mention taking the very expensive special train up to Jungfraujoch, which requires reservations. You don't need to make that part of the trip for excellent Alpine views, and it is both expensive and time consuming). His base point is Interlaken Ost, which is a gateway to the Lauterbrunnen Valley and the Jungfrau Region.
This panoramic map may help you decipher the complex transportation network near the Jungfrau:
www.swissholidayco.com /Public/Assets/User/fi les/Map-of-Jungfrauregion1.jpg
To get to Interlaken Ost from Basel requires approximately 2 hours. I have attached a screen grab of some morning journeys. Many do not require a change of trains.
I have also taken a screen grab from the detail obtained by clicking on one route, which does not require any changes of trains. You will note that it is an Intercity Train ("IC") not a regional train. Intercity trains permit reservations. To the right, you will see an "R." The footnote explains that the "R" means Reservations possible.
Although possible, it will not be necessary.
This useful page, as linked by Alpenrose, is quite helpful to understand the timetable:
Note that the fares displayed are approximately 1/2 the cost for those traveling without any sort of discount. Essentially all Swiss residents buy a "Half-Fare Card" (Halb-Tax) wich gives them a usually 50% discount on regular trains, buses, boats, cableways throughout Switzerland. You may be traveling without such a pass and will pay approximately double that price if you simply buy a normal ticket.
The timetable will let you figute out times for the various options mentioned by Arno.
In addition to the panoramic map linked above, this map is useful to get the lay of the land, and give you names of specific station stops, which can be entered into the timetable:
I have zoomed it in to zoom level 32, which makes the train lines clearly visible. Zoom out and you can see the relative location of the major cities.
In the menu, under "Points of Interest" select "Traffic" and you can turn on a layer with icons for the station stops. I turned on "Trains" and "Mountain Transport." Mouse over an icon and you get the official name for that station stop.