The pronunciation of Esperanto should remind one of that of Italian. To be more accurate that of Southern Italy, of Calabria, as well as that of the Slavic countries near Italy such as Serbo-Croatian and Slovenian, and also, to a lesser degree to that of Polish and of Russian near the Baltic.
English is a language especially distant phonetically from Esperanto, the general soundscape it fosters must not be relied upon, though, fortunately, nearly each individual sound of Esperanto is to be easily found is many English words of common American or Irish use, with the exception of the r which has to be learned again ideally, as nearly all Esperantists trill it like in Mediterranean or Slavic languages, and of the rough ĥ or double h which in both English and Esperanto has very little use outside proper nouns such as Loch, C'hutzpah, Czec'h.
What is most difficult for an anglophone to master as for the pronunciation of Esperanto is doing one's best to avoid the neutral vowel "shwa" of words such as burden, first, burglar, girl ... to which nearly all English vowels tend to in casual conversation when away from the word's main stress, as a in about, Florida, o in occur, forgive, today, e in behind, spoken, i in cabin, pumpkin, u in occur, fortune... There is no such sound in Esperanto nor are you supposed to let unstressed vowels slide into it out of the law of least effort. First try never to utter a shwa vowel when practising Esperanto. The vowels are clear-cut, exactly like all the other same letters, and perfectly distinguishable from those denoted by other letters : there is for instance no overlapping whatsoever of a and o like Americans commonly do (also Russians of Moscow) when pass from dog to car.
The five vowels of Esperanto are a very wide-open a as in car, ah! haha!,
a neat, constant, rather open e as in bet, there, bread (not closed and gliding from e to ee as in grey, break), a very sharp, constant and closed i as in machine, Marie (or like ee in keen, see.. ), a neat, constant, rather open o as in core, bought, rot (but never as open to be to close to a as the o of volume or lot, nor too closed and gliding from e to oo as the o of note) while u is always very closed-mouth as in rule, true or even better as oo in boom, boo, groove...
The a is supposed to be central, that is to say never as far back as in brawl and never as far front as in cat, bat... : the tongue is as far away from the roof of the mouth both at the front and rear. E is a rather open front vowel (the tongue is nears the palate in the front part of the mouth), never central as in driven or too open as in clerk. I is a front vowel, never central as in torrid, will. The o is clearly a back vowel, never too central as in obey, and so is the u : never too central as in July or occupy (many American Esperantists commit that error) and never gliding from y to oo as in due, cute. Avoiding glides in words where there are not is one of the most important exercises to be understood by other esperantists. Most of the vowels English-speakers pronounce when they see a single letter on paper are gildes or diphthongs when they don't tend to shwa's. Pure constant vowels are to be found in English as precious marks to rectify one's pronunciation but they are not the most frequent. Passing from English to Esperanto is like passing from a landscape composed mostly of greys, ochres, browns, army greens fading into each other to one where there are mostly rainbow colours that have neat contours as in a mosaic.
Zamenhof in his textbooks told his students to pronounce a moderately open e and o, but judging by the audio recordings he left and also by the authors who later on did their best to imitate him, as well as by the languages he admired most for opera and classical song, starting with his own native one which was Polish and ending with his favourite one which was southern or eastern Italian, these o's and e's as well the a's were of a very open-mouthed kind though always clearly distinguishable from each other (which people in Naples fail to do), while the i and u remain very closed.
However, even though the position of the tongue relatively to the mouth is clearly defined in a standard way, it is very explicitly allowed that that of the lips is subject to no constraint : most frequently front vowels are pronounced with the lips drawn sideways, as in the examples I have just provided above. But they can now and then be left relaxed, as with the e of pen and the i of pin or quickies, if that is your instinct for your speech to sound better to your own taste. When e is touching another vowel the lips tend to close more most naturally to distinguish it better from the other one resulting in a sound nearer greying, gay-art(but never like in great). Same thing for the o and u : they most normally are to be pronounced with the lips rounded but that is not an obligation. If the general harmony calls for it the o can be pronounced with the lips more relaxed like in dove, love (US GA pronunciation), and the u can be nearer that of put in the same way. When the o is touching another vowel the lips may close more naturally as in Joey, echo-art.
There are real diphthongs in Esperanto, though not as regularly appearing as in English : aŭ is nearly exactly like English ow in cow, mouth (resulting in browse being spelt braŭzi), aj is nearly exactly as English long i of pie, eye, glide... ej is strikingly similar to the English long gliding a of shake, face (resulting in Facebook being spelt Fejsbuko). Likewise oŭ is pronounced like Americans pronounce bowl (boŭlo in E-o) with a glide going from open o to oo, but by a strange choice from the speakers and word-coiners very few genuine words of E-o have adopted such a glide that is there mostly for proper nouns and brands. The diphthong oj is nearly exactly like its English counterpart (ĝoj' = joy both in sound and meaning) and is very frequent as it is the plural of all nouns. The diphthong uj is present in relatively few words but that are of very frequent use in particular among the plural pronouns, and there are only a tiny few examples of it in English to practise it unfortunately : gooey, ruin. The diphthong eŭ is quite present in Esperanto for scientific words of Greek origin : terapeŭtiko = therapeutic. Unfortunately it doesn't exist at all in General American but it is the most frequent Australian pronunciation (or Estuary British English) of long o as in no (ne-oo), alone (ë-le-oon). There are no iŭ's or ij's or uŭ's as of now that have been introduced for any Esperanto word not even a proper noun. Though it is permissible, it is especially not advised to pronounce e's and o's as closed ones in E-o, except when touching another vowel or half-vowel, because the diphthong ej is exactly like an English long closed e-sound glide and most Esperanto ears will pick up the ej diphthong when hearing closed e as in French coupé.
The consonants of E-o pose far less problems than vowels for English speakers to sound un-English in Esperanto. B is exactly the English b. P is exactly the English as in spin, spill but not as in pill, which Esperanto would write phil'. You must care to pronounce always the former unaspirated one when you meet with an Esperanto p. The d of Esperanto is dental like in bread or breadth, never retroflex like in dock. The t of Esperanto is always un-aspirated and dental as in stick, stool, never aspirated and retroflex like in tick tock. The hatted letters ĉ, ŝ, ĝ, ĵ correspond exactly and perfectly to English ch (or tch), sh, j (or dge), zh (or si as in vision). Only ĥ is problematic but as I told above it is used mostly for proper nouns in geography like the English ch of loch or Bach. There is no w in Esperanto. The letter is not part of the official alphabet. The sound of it exists but it is simply rendered by an ordinary vowel u at the beginning of a syllable : trotuaro, kuiri, tuito (English tweet). The half-vowel ŭ is too short to be used to represent the sound of an English w, which is as long as a full vowel, and moreover it always colours by merging, according to its own definition, the next vowel which E-o considers to be desirable only to form diphthongs at the end of a syllable : if you write ŭa you will get something very near the short u of but or Trump which is too ambiguous a sound to be any use for Esperanto : therefore avoid to write it in E-o altogether. Most words that have a w sound in English are transcribed with a v, not a ŭ, especially between a consonant and a vowel. Twitter is thus transcribed tvitero and quality kvalito. In such a position, v represents, rather than always a fricative v, a very short w sound like in French suite, fuite or a lazy English pronunciation of sweet : this prevents the w sound from eating the length of a whole syllable when transcribed in E-o. The other consonants are pronounced exactly as in English and call for no remark. C is just ts as bits or schmaltzy. But the sound of dz has no separate single letter. ng can be almost like in English be pronounced either with a full g or without it but only with the modified n of song if there is no vowel following it. There is no English th sound in Esperanto but it may be approximated with tf and dv surprisingly well.