For all those we can say that the prefix is fossilized in some words, and it's meaning almost opaque, but we can deduce a set of meanings from other examples and the prefixes are productive.
be- is comparable to English be-, cp. become, bewitched and roughly denotes direction of activity; cp. bei, by, Lat. com "with", e.g. correct - "berichtigen". Cp. e.g. a neologism from "to twitter", hence * "jemanden betwittern" would be borrowing the semantics of belabern (to talk an ear full), or pershaps behandlend (to deal with).
er- is ... I haven't thought about it before. It's relatively rare, compared to the others. I'd guess that it's akin to her-, like hin- to in; on the model of erhalten (cp. syn. einnehmen), ergeben (cp. syn. aufgeben). For Ergebniss "result", I would guess, since I don't know any better, that Ergebnis is akin to Latin ergo (thus, consequently), which is from ex+rego, of the same root as rechnen and to reckon; Thus cp. errechnen.
ver- is a conflation of many different roots, for sure. Which is which is hard to tell for a common native speaker. cp Latin per, Ger. vor, für, and more, note the bunch of translations containing pro- below. There's a negative connotation in many words, e.g. vergebens "in vein", verrechnen "miscalculate", verführen "to lure, deceive", etc. but in some words, especially verletzen (to hurt, damage) and verlieren (to lose, cp. forlorn), verlassen "to leave", the negative connotation is already in the stem (all relate to the stem in En. loss, less), and the common notion of the prefix is mutation, change of state, cp. perlocate, permutate, etc. (more or less with a sense for, towards).
It also matters what stem the prefix is attached to. With nouns, the meaning is rather clear. "be--en" means the object is acting onto something (benebelt, foggy mind from smoke), with er- I can't think of a single example, and with ver- it means to turn into, or apply it to. The difference is often not clear, as e.g. golden (viz vergolden "to apply a gold finish") might be adjectival, and nebeln verbal, prefix or not. This ambiguity in the suffixes (morphemes are not real suffixes, but oh well) or rather the versatility of the stems may be part of the reason that it works well. On the other hand, we also have Geltung "dignity, validity", Vergeltung "retaliation" but gelten "to be valid", vergelten "to retaliate"
A few examples:
suchen "to seek"
besuchen - to pay a visit (to someone)
ersuchen - to seek help (in[?] someone)
versuchen - to try out
aussuchen - to pick out, choose
heraussuchen - to find out
steuern "to stear", Steuer "tax"
besteuern - to ask tax, set-up a tax (for something)
versteuern - to pay tax (**for* something)
Rechnen "to calculate" [to sum up, collect, cp. Rechen "rake"?]
mostly synonymous ausrechnen, errechnen, berechnen, if talking about math homework. The difference is in the inflected forms
berechnend - calculating, scheeming [highlights the aim, the subordinate nature of the process; to achieve by calculating]
errechnen - to calculate a result [chiefly highlights the achievement]
*verrechnen - 1. to count two calculations against each other (e.g. liabilities versus outstanding income with a single partner, also aufrechnen, lit. to sum up), 2. to miscalculate [I could see 1 and 2 relate a bit as working untidy instead of "doppelte Buchführung", but deem it unlikely].
Abrechnung - billing [ab "off"; denotes that the calculation the conclusion of a bigger process (of business in most cases]
incidentally, rechnen, En. reckon, Proto-Germanic *rekanōną "to count, explain", from Proto-Germanic *rekanaz (“swift, ready, prompt”), is from Proto-Indo-European *h₃reǵ-, just as ergo, viz Ergebnis "Result", Tag der Abrechnung "day of reckoning".
führen "to guide, lead"
hinführen - lead to
herführen - bring here
entführen - take away, kidnap; lead astray
verführen - to lure, deceive
befürworten - to concur, endorse, favor, support [this is fossilized, befür- exists only in this one idiom, as far as I know, and no analog that the be- prefix attaches to is in use, though Fürwort could be derived as "endorsement"; A *Führwort could be searched in Geleitwort, Leitwort, Vorwort; Just my two cents]
no er + führen exists. But we have Erfahrung "experience, perception".
fahren "to drive, go"
befahren, die weniger befahrene Straße** - the road less traveled
erfahren - 1. to experience 2. analogue to erlaufen "to achieve something by walking the distance"
verfahren - 1. to process, conduct 2. to go the wrong way
The funny bit is, that fahren derives from the per root. I'd like to compare purport, at another time though.
halten "to hold"
behalten - to keep [to hold by; chiefly behalte bei ...]
erhalten - to attain [to take in]
verhalten - to behave (somehow) ["behave!" would need an object for verhalte dich ...! or it would be e.g. benimm dich instead, viz Benehmen.
greifen "to grab, grasp"
begreifen - to take in, understand [cp. Dutch begripjen, Ger. Grips "smarts"; Ger. Begriff "word, meaning"]
ergreifen - to apprehend (a fugitive) [to take them in, also festnehmen, feststellen, but fest stellen also means to find out a fact]
ergriffen - moved, taken away
vergreifen - to mistake, to take a wrong (thing, tone "im Ton vergreifen")
vergriffen - out of stock
cp. Gr. grapho "write, sign, draw" for this one.
schreiben "write", Schrift "writing"
beschreiben - describe
Einschreiben - a letter send in
verschreiben - 1. (med.) to ascribe medication or treatment [lit. to give a receipt for meds] 2. to make a mistake in writing, a typo
verschriftlichen - to set in writing
tragen "to carry"
Betragen - behaviour, to carry oneself [possibly calqued of Lat. cura "care"?]
Ertrag - yield
Vetrag - contract; vertragen - 1. to get along with each other 2. (col.) to move (a bunch)
vortragen - to perform, to bring before
eintrage - to list, matriculate, to write down into ...
austragen - the delist
Antrag - proposal, request
[again, cp. *Hreg? ragen, hervorragend, reach etc.?]
sprechen "to speak"
besprechen - bespeak, negotiate, debate, to go over (this); Besprechung - meeting, critique
versprechen - 1. to promise 2. to misspeak
handeln "to handle, to act, to deal"