mano pubbaṅgamā dhammā manoseṭṭhā manomayā
manasa ce paduṭṭhena bhāsati vā karoti vā
tato naṃ dukkham anveti cakkaṃ va vahato padam

mano pubbaṅgamā dhammā mansoeṭṭhā manomayā
manasa ce pasannena bhāsati vā karoti vā
tato naṃ sukham anveti chāyā va anapāyinī

Mind precedes experience, mind is foremost, [experience is] mind-made.
If, with a corrupt mind one speaks or acts:
From this disappointment and suffering follow as the wheel, the foot of the ox.

Mind precedes experience, mind is foremost, [experience is] mind-made.
If, with a clear mind one speaks or acts:
From this happiness and well-being follow like an inseparable shadow. This is a fairly literal translation which largely retains the structure of the Pāli. Two interesting philological features are pointed out by K.R. Norman and John Brough with regard to these verses. Firstly the word anveti appears to be a Sanskritisation. Norman suggests that Pāli would usually resolve the consonant cluster nv to nuv, but here it doesn't. Secondly Brough (p.243) points out in his notes to the Gāndārī Dhammapada that vahatu (vahato being the genitive case) is an archaic word not in the Pali Text Society's Pali-English Dictionary. The word means ox, and vahato padam is “foot of the ox”, where pada is also used in it’s archaic sense of "foot", as opposed to it's later more abstract meaning "word" as in the title of the Dhammapada: "the word of the doctrine".
The archaic forms suggest age, but the Sanskritisation may also indicate later editing or composition. Perhaps an old image re-used? Interestingly the Pāli commentators (Sri Lanka, ca 5th century) seem to have understood the sense of “vahato padam” but not the words, so come to the right conclusion by some tortuous arguments. This is evident in many Dhammapada translations which treat vahato as a present-participle meaning something like "bearer".
In these verses the terms mano and dhammā (nominative plural) are twinned, as are sukha and dukkha, and paduṭṭha and pasanna. Mano and dhammā in this context are the mind which senses mental "objects", and those "objects" or dhammas. This is the more specific meaning of mano, which is sometimes also used synonymously with other words for "mind" such as citta and viññana. Dhamma has such a wide range of meanings that it can be misleading to settle on one in particular, but here does seem to indicate the mental phenomena which the mind senses - in these cases it is usually written in lower-case. Note also that mind here includes the emotions, and other subjective experiences. Mano also coordinates the mental responses to the information coming in from the five physical senses. I have chosen the word "experience" as a translation of dhamma in this case because it covers both the mental and physical aspects. I have justified using "experience" to translate dhamma in other contexts as well, particularly in my essay on the Buddha's Last Words.
Mano is that part of us that cognizes experience, that part of us that knows we are experiencing. And these verses are saying that mano comes first. But why? Other Buddhist models of the psyche suggest that mind, in the sense of citta, arises in dependence on contact of sense organ with sense object. It is important not to get caught up in the various models here. This is a pragmatic teaching. Mind comes first in these verses because mental states determine actions, and therefore consequences. The Buddha famously said: Cetanāhaṃ, bhikkhave, kammaṃ vadāmi - I say, monks, that intention is action [A vi.63]. This is why we must focus on the mind.
Actions arising from a mind which is paduṭṭha (spoiled, rotten, corrupt, literally "made bad") lead to dukkha - which I translate as "disappointment". However actions which are directed by a mind which is passana (clear, bright, good) result in sukha or bliss. The latter term is one which was in use before the Buddha. Brahman, the universal absolute has three characteristics: being, consciousness, and bliss (sukha). Brahman is also nitya (Pali nicca) or eternal. So in a Brahminical context sukha has a connotation of the goal of the spiritual life: union with Brahman. I think the Buddha may well have been employing sukha as a synonym for nibbāna - drawing on the Brahminical imagery as we know him to have done in many other cases. Despite this I have translated sukha as happiness and well-being. References to underlying Brahminical metaphors are often confusing to modern Buddhists who are frequently ignorant of the Brahminical context of some of the Buddha's sermons. Happiness here though does mean true happiness, the highest happiness, the bliss of nibbāna.
Dukkha, then, is the opposite of nibbāna. This gives it a much broader scope than is usually suggested by translations such as "suffering". When we read "suffering" we tend to think of physical pain or injury. But dukkha characterises all unenlightened experience. At this point you may be thinking "aha! Jayarava has fallen into that old trap of stating, contra the Dharma itself, that everything is suffering". However I am making a more subtle point. Not every experience is physically painful, but we the unenlightened have habitual tendencies which make even pleasure a disappointment. This operates at the level of immediate responses to vedana or sensations. Typically when we experience a pleasant sensation we want it to last, and when we have an unpleasant sensation we want it to stop now. We seek out pleasure, and avoid pain. I have argued at length in my essay on the Buddha's Last words that it is at this level of experience that dependent arising is really important. It is experiences (sensations including mental sensations and our responses to physical sensations) that are impermanent (anicca). The disappointment (dukkha) comes because we fail to grasp the nature of experience - we think of it as, or desire it to be, lasting (nicca). Because experiences (dhammā) are impermanent (anicca) they are disappointing (dukkha). The argument showing how this ties in with the doctrine of anattā would take a bit long to spell out, but it relates to the Brahminical idea about ātman being Brahman in the microcosm, and therefore necessarily having sukha as a characteristic - anything which has dukkha as a characteristic cannot be the ātman. A purely psychological understanding of ātman as simply 'ego' is, I think, a bit limited.
The verses are saying that we experience dukkha if our mind is corrupt. That is, if we fail to see and understand the nature of experience (yathābhūta-ñāṇadassana), then suffering follows, just as the wheel follows the ox which draws the cart - as the wheel follows the foot in the simile of the verses. The image for sukha is subtly different. If our mind is clear and bright (pasanna) then we see things as they are, and bliss cleaves to us. For dukkha the sense is that one thing follows another, and the two are distinct. A shadow however is simply an extension of our body - the shadow moves with us, moves as we move, instantaneously. Our shadow is inseparable (anapāyinī literally not-going-away). Apāyinī can also connote "a falling away (in conduct)" or "transient state of loss or woe after death" [PED sv apāya] so that anapāyinī (not-apāya), like sukha, suggests the goal of the Buddhist life: not falling away from good conduct, not falling into state of loss or woe.
So mind is first, mind is foremost, and things are said to be mind-made because mind determines the results of our actions. Those results are experienced as vedana (pleasant, unpleasant, or neutral sensations) and it is our response to vedana that determines whether we experience 'being' as dukkha or sukha, ie saṃsāra or nibbāna. If we fail to understand our existential situation we can expect only dukkha. Of course Buddhism offers us a plethora of tools for the job of waking up, and tells us that everyone can wake up. So while unawakened experience is disappointing, it is not the only possibility. There is every reason for optimism.

~~oOo~~
References
Brough, John, ed. The Gandhari Dharmapada (London: Oxford University Press, 1962). Norman, K.R., trans. The Word of the Doctrine (Oxford: The Pali Text Society, 1997). See also my commentary on Dhammapada v.3-4