For those who prefer texts in Slavonic and Russian, the Moscow Patriarchate has been posting reader service texts for both the Vigil and Typika for Sundays and feasts, linked at the top of their official page: http://www.patriarchia.ru/
For those who want texts in English, however, here is the next set of texts:
The Eves
For the Eves of the upcoming Sundays and Feasts, you could ideally do the Vigil. The fixed portions can be downloaded here:
http://www.saintjonah.org/services/reader_vigil.doc
or viewed in HTML, here:
http://www.saintjonah.org/services/vigil.htm
For the Rubrics, see: http://www.saintjonah.org/rub/
The variable portions of the service can be downloaded here (all of these would be served on the eve of their respective days):
For the Sunday of the Holy Fathers (May 31st):
http://www.saintjonah.org/services/pascha6.doc
For the Sunday of Pentecost (June 7th):
http://www.saintjonah.org/services/vigil_pentecost.doc
For the Sunday of All Saints (June 14th):
http://www.saintjonah.org/services/pent1.doc
For the Sunday of All Saints of Russia (June 21st):
http://www.saintjonah.org/services/pent2.doc
The rest of these services require two files, because these combinations do not repeat annually. In addition to the files linked for the Sundays below, you will need to use the appropriate Katavasia, which for this time period is the Katavasia of the Theotokos. Also, there are some hymns that are appointed according to which Matins Gospel is read. To find out which one is read, you need to look at the Rubrics. For those texts, you will find them here: http://www.saintjonah.org/services/matinsgospel.doc Those hymns are usually done at the Exapostilaria and then at the Doxasticon at the Praises.
For the 3rd Sunday after Pentecost / Prophet Amos (June 28th):
http://www.saintjonah.org/services/6-15_prophet_amos.doc
combined with the Octoechos:
http://www.saintjonah.org/services/tone2.doc
For St. John of Shanghai (Saturday, July 4th) (Octoechos text included):
http://www.saintjonah.org/services/stjohnofshanghai_t2.doc
For the 4th Sunday after Pentecost / Hieromartyr Eusebius, Bishop of Samosata (July 5th):
http://www.saintjonah.org/services/6-22_hm_eusebius.doc
combined with the Octoechos:
http://www.saintjonah.org/services/tone3.doc
For the Nativity of St. John the Baptist (Tuesday, July 7th) (no Octoechos text needed for this service):
http://www.saintjonah.org/services/vigil_nativity_forerunner.doc
For the 5th Sunday after Pentecost / Ss. Peter and Paul (July 12th):
http://www.saintjonah.org/services/vigil_sspeter&paul_sun.doc
combined with the Octoechos:
http://www.saintjonah.org/services/tone4.doc
However, if doing Vigil is too much for you at present, you could do Small Compline, and take the canon of each of the above days, and read it immediately after the Creed, and then repeat the Kontakion that is appointed after Ode 6th of the canon after the following Trisagion.
Typika
In place of the Liturgies, you would do Typika:
For the Sunday of the Holy Fathers (May 31st):
http://www.saintjonah.org/services/typika_pascha6.doc
For the Sunday of Pentecost (June 7th):
http://www.saintjonah.org/services/typika_pentecost.doc
For the Sunday of All Saints (June 14th):
http://www.saintjonah.org/services/typika_pent1.doc
For the Sunday of All Saints of Russia (June 21st):
http://www.saintjonah.org/services/typika_pent2.doc
For the 3rd Sunday after Pentecost / Prophet Amos (June 28th):
http://www.saintjonah.org/services/typika_pent3.doc
For St. John of Shanghai (Saturday, July 4th):
http://www.saintjonah.org/services/typika_stjohnofshanghai_t2.doc
For the 4th Sunday after Pentecost / Hieromartyr Eusebius, Bishop of Samosata (July 5th):
http://www.saintjonah.org/services/typika_pent4.doc
For the Nativity of St. John the Baptist (Tuesday, July 7th):
http://www.saintjonah.org/services/typika_nativity_forerunner.doc
For the 5th Sunday after Pentecost / Ss. Peter and Paul (July 12th):
http://www.saintjonah.org/services/typika_pent5.doc