OP mentioned Dillon and Lee specifically and I guess I'd opine they are, more or less, near the opposite ends of the spectrum at least with respect to cost. I've used both along with Redding and RCBS and a myriad of just about every brand of die, scale, powder measure, etc.
If you are just starting out it makes sense, IMHO, to begin with a single-stage press. It's economical, simple, and doing the various steps separately will furnish a real feel for the process. Eventually, if you like reloading, you may wish to upgrade to something a bit faster - but progressive and multi-stage press setups won't load any more accurate ammo than a single stage. If you end up reloading for rifle a single stage may always be something warranting bench space.
As far as brands go you will, no doubt, find as many opinions as there are readers of your post. In all candor there is not as much difference between them as some would have you believe. Lee is likely the most economical if purchased new. I've used Lee products for many, many years and have had excellent results. Pertaining to single stage presses I admit to being partial to my Redding Boss press but it's orders of magnitude more costly than Lee and does not produce a finished product that is distinguishable from cartridges made on my Lee equipment. A Lee turret press may be a good compromise as it can be used as a single stage and can enable you to go much faster once you get the hang of it. Mine has preformed splendidly for many years.
You may want to check out used presses and dies via ebay or gunbroker as some good deals can occasionally be had. I've found some really nice older dies and beam scales on ebay and they have served very well.
Good luck and have fun!